# User manual | Morrison - Three Problems in Nonlinear Dynamics with 2_1 Parametric Excitation - 2006.pdf ```THREE PROBLEMS IN NONLINEAR DYNAMICS
WITH 2:1 PARAMETRIC EXCITATION
A Dissertation
Presented to the Faculty of the Graduate School
of Cornell University
in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy
by
Tina Marie Morrison
May 2006
c 2006 Tina Marie Morrison
ALL RIGHTS RESERVED
THREE PROBLEMS IN NONLINEAR DYNAMICS
WITH 2:1 PARAMETRIC EXCITATION
Tina Marie Morrison, Ph.D.
Cornell University 2006
Parametric excitation is epitomized by the Mathieu equation, ẍ+(δ +ε cos t)x = 0,
which involves the characteristic feature of 2:1 resonance. This thesis investigates
three generalizations of the Mathieu equation:
1) the effect of combining 2:1 and 1:1 parametric drivers:
ẍ + (δ + ε cos t + ε cos ωt)x = 0
2) the effect of combining parametric excitation near a Hopf bifurcation:
ẍ + (δ + ε cos t)x + εAẋ + ε(β1x3 + β2x2 ẋ + β3 xẋ2 + β4ẋ3 ) = 0
3) the effect of combining delay with cubic nonlinearity:
ẍ + (δ + ε cos t)x + εγx3 = εβx(t − T )
Chapter 3 examines the first of these systems in the neighborhood of 2:1:1
resonance. The method of multiple time scales is used including terms of O(ε2 )
with three time scales. By comparing our results with those of a previous work on
2:2:1 resonance, we are able to approximate scaling factors which determine the
size of the instability regions as we move from one resonance to another in the δ-ω
plane.
Chapter 4 treats the second system which involves the parametric excitation of
a Hopf bifurcation. The slow flow obtained from a perturbation method is investigated analytically and numerically. A wide variety of bifurcations are observed,
including pitchforks, saddle-nodes, Hopfs, limit cycle folds, symmetry-breaking,
homoclinic and heteroclinic bifurcations. Approximate analytic expressions for bifurcation curves are obtained using a variety of methods, including normal forms.
We show that for large positive damping, the origin is stable, whereas for large
negative damping, a quasiperiodic behavior occurs. These two steady states are
connected by a complicated series of bifurcations which occur as the damping is
varied.
Chapter 5 examines the third system listed. Three different types of phenomenon are combined in this system: 2:1 parametric excitation, cubic nonlinearity, and delay. The method of averaging is used to obtain a slow flow which
is analyzed for stability and bifurcations. We show that certain combinations of
the delay parameters β and T cause the 2:1 instability region in the δ-ε plane to
become significantly smaller, and in some cases to disappear. We also show that
the delay term behaves like effective damping, adding dissipation to a conservative
system.
BIOGRAPHICAL SKETCH
“I am what I am at this moment, not what I was and certainly
not all that I shall be.”
– C. S. Wyatt
Tina Marie Morrison was born in 1978 to a loving mother in New Britain, CT. She
grew up with her two younger siblings, Dwight Jr. and Nicole. Tina graduated
from E.C. Goodwin Vocational High School in New Britain and received a degree in
Mechanical Drafting. Her shop instructor, Fred Norton encouraged her to attend
college and pursue a career in Mechanical Engineering.
In 1996, Tina entered the Mechanical Engineering program at the University of
Connecticut, Storrs. She received her bachelor’s degree along with a minor degree
in Applied Mathematics. Upon graduation, the Mechanical Engineering department invited her to participate in their Accelerated One Year Master’s program.
She completed her master’s with a concentration in Nonlinear Vibrations under
the advising of Dr. Kevin Murphy.
In the spring of 2002, Tina was accepted into the doctoral program in the Theoretical and Applied Mechanics department at Cornell University. The following
year, she began her research training in Nonlinear Dynamics under the guidance of
Dr. Richard Rand. She received her doctorate in 2006, with a minor concentration
in Applied Mathematics. Tina is the first in her family to graduate college and
receive a Ph.D.
iii
To my mother, Diane
For your love, encouragement and sacrifice
iv
ACKNOWLEDGEMENTS
“Reflect upon your present blessings, of which every man has plenty,
not on your past misfortunes, of which all men have some.”
– Charles Dickens
During my years as a doctoral candidate in Theoretical and Applied Mechanics, I
received tremendous support and assistance from a number of individuals.
Foremost, I express my sincere appreciation to Richard Rand for being an
excellent thesis advisor and teaching mentor. He worked closely with me to ensure
my progress and success in the program. Even when I moved to Boston for a year,
his support and guidance did not waiver. During my time at Cornell, Richard
encouraged excellence in my research and affirmed my abilities as a teacher. For
this, I am truly grateful.
Upon completing my first semester at Cornell, Steve Strogatz recognized my
enthusiasm and commitment to the IGERT program and offered me an IGERT
fellowship for my second year. I am appreciative of his support and encouragement
during my candidacy years. In addition, I thank Alan Zehnder for carefully reading
and commenting on my thesis.
Moreover, I am grateful to four very special women for being extremely helpful
and supportive: Cindy Twardokus, Polly Marion, Dolores Pendell and Sreemati
Mukherjee. In addition, I am thankful to Tim Healey for providing career advise
and for assuring me of my place in the T&AM family.
Likewise, I am pleased to have shared my graduate experience with Geoff Recktenwald. During the course of our studies, he and I discussed research, philosophy,
and world issues. We enjoyed many moments of fun and laughter. In addition,
v
I am grateful for the moments we shared with our officemates, Carlos Torre and
Dennis Yang, and with the talented T&AM 2002 crew: Bev Thurber, Hui Chen,
Tuhin Sahai, Sovan Das and Danny Abrams. I am also thankful to Mike Stubna
and Tamás Kalmár-Nagy for answering my questions about thesis preparation and
research.
Finally, the completion of this thesis would not have been possible without the
constant encouragement from my friends and family. I am deeply appreciative
to my family for their unconditional love and support; to Jennifer Brown for her
inspiration and friendship; to Carla Wiese for her outrageous free spirit; to Angela
Wilson, Nameste; to Danielle Lyles for opening her home to me during my trips
from Boston to Ithaca; to Rodolfo Cuevas para su amistad; Edoardo Carta for
creating a home away from home; and to Ofer, ani ohevet otcha.
vi
TABLE OF CONTENTS
1 Introduction
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2
3
2 Parametric Excitation
6
2.1 Mass-Spring Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Pendulum with Moving Support . . . . . . . . . . . . . . . . . . . . 9
2.3 The Mathieu Equation . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 2:1:1 Resonance in a Quasiperiodic Mathieu Equation
3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Method of Multiple Time Scales . . . . . . . . . . . . . .
3.4 Analysis of the Slow Flow . . . . . . . . . . . . . . . . .
3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . .
4 2:1
4.1
4.2
4.3
4.4
Resonance Near a Hopf Bifurcation
Motivation . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . .
Two-Variable Expansion Method . . . .
Analysis of Slow Flow . . . . . . . . . .
4.4.1 Equilibria . . . . . . . . . . . . .
4.4.2 Limit Cycles . . . . . . . . . . . .
4.5 One Parameter Bifurcations . . . . . . .
4.6 Examples . . . . . . . . . . . . . . . . .
4.6.1 Example One . . . . . . . . . . .
4.6.2 Example Two . . . . . . . . . . .
4.6.3 Example Three . . . . . . . . . .
4.7 Two Parameter Bifurcations . . . . . . .
4.7.1 Hamiltonian System . . . . . . .
4.7.2 Preserving Closed Orbits . . . . .
4.8 Symmetry Breaking Bifurcation . . . . .
4.9 Homoclinic Bifurcation . . . . . . . . . .
4.9.1 Double Zero Eigenvalue . . . . .
4.9.2 Homoclinic Bifurcation Revisited
4.10 Discussion . . . . . . . . . . . . . . . . .
4.11 Conclusions . . . . . . . . . . . . . . . .
vii
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14
15
15
20
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25
28
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29
30
30
31
33
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36
38
40
42
44
47
50
50
57
60
63
64
71
75
77
5 2:1 Resonance of a Delayed Nonlinear Mathieu Equation
5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 The Hsu-Bhatt Equation . . . . . . . . . . . . . . . .
5.2.2 The Nonlinear Hsu-Bhatt Equation . . . . . . . . . .
5.3 First Order Averaging Method . . . . . . . . . . . . . . . . .
5.4 Analysis of the Slow Flow . . . . . . . . . . . . . . . . . . .
5.4.1 Slow Flow Equilibria: Stability and Bifurcation . . .
5.4.2 Example 1 . . . . . . . . . . . . . . . . . . . . . . . .
5.4.3 Example 2 . . . . . . . . . . . . . . . . . . . . . . . .
5.5 Degenerate Hopf Bifurcation . . . . . . . . . . . . . . . . . .
5.5.1 Order ε Delay . . . . . . . . . . . . . . . . . . . . . .
5.5.2 Order 1 Delay . . . . . . . . . . . . . . . . . . . . . .
5.5.3 Numerical Investigation . . . . . . . . . . . . . . . .
5.6 Comparison with Numerical Integration . . . . . . . . . . . .
5.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . .
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79
80
80
83
86
89
91
92
95
99
103
103
105
108
108
112
113
6 Conclusions and Future Work
6.1 The Duffing Oscillator . . . . . . . . . . .
6.1.1 Elliptic Function Solution . . . . .
6.2 The Forced Duffing Oscillator . . . . . . .
6.3 The Duffing Oscillator with 2:1 Parametric
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116
117
119
120
126
References
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Excitation .
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127
viii
LIST OF FIGURES
2.1
2.2
2.3
2.4
3.1
3.2
3.3
3.4
3.5
3.6
4.1
Mass-spring oscillators:
(a) simple harmonic oscillator with natural
√
frequency of k1 + k2 , with m = 1, (b) simple harmonic oscillator
with an external forcing function, F (t) = A cos Ωt, and (c) simple
harmonic oscillator with time-varying stiffness k2 = k2 cos ωt. . . . 8
The pendulum: (a) simple pendulum with a moving support, F (t) =
F cos ωt (b) inverted pendulum with moving support. . . . . . . . 10
Transition curves for the Mathieu equation for ε < 1 which separate
regions of stability (gray) from regions of instability (white). . . . 12
Transition curves for the inverted pendulum examples. The dot a
corresponds to a stable θ = 0; the dot b corresponds to an unstable
θ = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Stability chart for the Mathieu equation ε < 1. The lowest order
equation is given for the instability tongue. U is unstable, S is stable.
Stability chart for the quasiperiodic Mathieu equation ε < 1. The
lowest order equations are given for the instability tongues. U is
unstable, S is stable. . . . . . . . . . . . . . . . . . . . . . . . . .
Stability chart for the quasiperiodic Mathieu equation in the δ-ω
plane for ε = 0.1. δ2:1 and δω:1 are the equations for the instability
tongues given in Figure 3.2. This figure is based on the lowest order
analytic approximation of Figure 3.2, in contrast with Figure 3.4
which is based on the numerical integration of equation (3.1). U is
unstable, S is stable. . . . . . . . . . . . . . . . . . . . . . . . . .
Stability diagram of equation (3.1) from Zounes, . This figure
is based on the numerical integration of equation (3.1), in contrast
with Figure 3.3, which is based on the lowest order analytic approximation of Figure 3.2. White regions are unstable, black regions are
stable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Transition curves near δ = 0.25 and ω = 1 based on equation (3.2)
from . Compare with Figure 3.4.
. . . . . . . . . . . . . . . .
Transition curves near δ = 0.25 and ω = 0.5 based on equation
(30). Also shown for comparison are the transition curves near
δ = 0.25 and ω = 1 based on equation (3.2) from . Compare
with Figures 3.4 and 3.5. . . . . . . . . . . . . . . . . . . . . . . .
16
16
17
18
19
26
Analytical bifurcation set in the A-κ parameter plane for Λ = −1/2
and B = 1. A is linear damping and κ = k1 /A, where k1 is detuning. P1 is a saddle-node, P2 and P3 are pitchforks and H is a Hopf.
These bifurcation curves are given by (4.20) and (4.27). Point Q
is the location where the saddle-node and Hopf bifurcations meet
tangentially. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
ix
4.2
Bifurcation set in the A-κ parameter plane for Λ = −1/2 and B =
1. The solid lines are the bifurcation curves found numerically,
while the dashed were found analytically. P1 is a saddle-node, P2
and P3 are pitchforks, H is a Hopf, LCF is a limit cycle fold,
SB is a symmetry-breaking, HoC is a homoclinic and HeC is a
heteroclinic bifurcation. The numbers on the RHS, (1), (2) and (3)
reference the location of κ used in Example 1, 2 and 3. . . . . . .
4.3 Phase portraits of the slow flow in the (u, v) plane for Example
One with κ = −0.80. Starting with positive damping, a pitchfork
bifurcation occurs at A, followed by a second pitchfork bifurcation
at B, a heteroclinic bifurcation at C and ending with a saddle-node
bifurcation at D. . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Caption for Example Two, κ = −0.35 begins on the next page.
.
4.5 Phase portraits of the slow flow in the (u, v) plane for Example
Three with κ = 0. Starting with positive damping, the initial flow
is the same as Example One where a pitchfork bifurcation occurs
at A, followed by a limit cycle fold at B, a symmetry-breaking
bifurcation at C, a subcritical Hopf at D, a second pitchfork at E,
and ending with a saddle-node bifurcation at F . The flow at  is
the same in Figure 4.4. . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Separatrix loop for the Hamiltonian System where κA < B2 .
. . .
4.7 Separatrices for the Hamiltonian System where κA > B2 . . . . . .
4.8 Numerical homoclinic orbits for the slow flow system plotted in the
u-v plane. (a) the separatrix in the conservative system, (b) the
destruction of the separatrix when the nonconservative terms are
added. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.9 Bifurcation set in the A-κ parameter plane for Λ = −1/2 and B =
1. P1 is a saddle-node, P2 and P3 are pitchforks, H is a Hopf,
LCF is a limit cycle fold, HoC is a homoclinic and HeC is a
heteroclinic bifurcation. SBn is a symmetry-breaking bifurcation
curve found by numerical integration and SBa is the symmetrybreaking bifurcation curve found analytically. . . . . . √
. . . . . .
4.10 Phase portrait of the slow flow at κ = −3/4 and A = −2/ 5. The
first transformation is moving the system from the origin to the
. . . . . .
location of the singular slow flow equilibria at (u1, v1).
4.11 Phase portrait of the Takens-Bogdanov normal form. . . . . . . .
4.12 Bifurcation set for Λ = −1/2. Point Q is the location where the
double zero eigenvalue occurs. P1 is the saddle-node bifurcation, H
is the Hopf bifurcation, HoCn is the homoclinic bifurcation curve
found by numerical integration and HoCa is the homoclinic curve
found by unfolding point Q. . . . . . . . . . . . . . . . . . . . . .
5.1
Stability chart for the Mathieu equation without delay, equation
(5.1) with b = c = 0. The solution is stable in the shaded regions.
x
41
43
45
49
53
54
55
62
66
72
76
81
5.2
Stability chart for the Hsu-Bhatt DDE, equation (5.1) with a =
c = 0 and T = 2π. The solution is stable in the shaded regions. .
5.3 Stability chart for the delayed Mathieu equation with T = 2π,
a = 1 and c = 0, . The solutions in the shaded regions are stable.
The darker shaded triangles are those in Figure 5.2. . . . . . . . .
5.4 Stability charts for Hsu-Bhatt equation (5.2) with T=π (top) and
T=2π (bottom). The shaded regions are stable. . . . . . . . . . .
5.5 Stability of the origin for Example 1, β = 14 and α = γ = 1. Black
is stable, white is unstable. The transition curves separating the
stability regions are given by equation (5.60) and by T = 2π. . . .
5.6 Bifurcations in slow flow equilibria for Example 1, β = 14 and
α = γ = 1. The letters a, b, c, d, e, f , g, h, i correspond to the
qualitative phase portraits shown in Figure 5.7. Pitchfork bifurcations occur on the Det(J ) = 0 curves, given by equation (5.60).
A change of stability occurs on the T r(J ) = 0 curve, T = 2π, in
the case that Det(J ) > 0 (non-saddle equilibria). No limit cycles
are born as we cross the latter curve because the associated Hopf
bifurcation is degenerate.
. . . . . . . . . . . . . . . . . . . . . .
5.7 Qualitative phase portraits for Example 1, β = 14 and α = γ = 1.
The letters a, b, c, d, e, f , g, h, i correspond to various parameter
combinations of δ1 and T as shown in Figure 5.6. The bifurcations from a to g, b to h, and c to i, respectively, are pitchforks.
The bifurcations from a to c, d to f , and g to i, respectively, are
degenerate Hopfs.
. . . . . . . . . . . . . . . . . . . . . . . . . .
5.8 Stability of the origin for Example 2, β = 35 and α = γ = 1. Black
is stable, white is unstable. The transition curves separating the
stability regions are given by equation (5.61) and by T = 2π. . . .
5.9 Bifurcation set for the slow flow equilibria for Example 2, β = 35
and α = γ = 1. The letters a, b, c, d, e, f , g, h, i correspond to
the qualitative phase portraits shown in Figure 5.7. The letters j,
k, l, m, n, o correspond to the qualitative phase portraits shown
in Figure 5.10. Pitchfork bifurcations occur on the Det(J ) = 0
curves, given by equation (5.61). Saddle-node bifurcations occur
on the Disc = 0 curves, given by | sin(T /2)| = 5/6. A change of
stability occurs on the T r(J ) = 0 curve, T = 2π, in the case that
Det(J ) > 0 (non-saddle equilibria). No limit cycles are observed
to be born as we cross the latter curve because the associated Hopf
bifurcation is degenerate.
. . . . . . . . . . . . . . . . . . . . . .
5.10 Qualitative phase portraits for Example 2, β = 35 and α = γ =
1. The letters j, k, l, m, n, o correspond to various parameter
combinations of δ1 and T as shown in Figure 5.9. The bifurcations
from j to l and from m to o are saddle-nodes. . . . . . . . . . . .
xi
81
82
85
96
97
98
100
101
102
5.11 Stability of the origin for β = 0.25 and α = 1. Comparison between analytical result based on slow flow (upper) with numerical
integration of equation (5.31) for ε = 0.05 (lower). Black is stable,
white is unstable. . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.12 Stability of the origin for β = 0.60 and α = 1. Comparison between analytical result based on slow flow (upper) with numerical
integration of equation (5.31) for ε = 0.05 (lower). Black is stable,
white is unstable. . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.13 The 2:1 instability tongue (5.80), for β = 35 , α = γ = 1. U is
unstable, S is stable. The white instability region lying in the plane
ε = 0.05 corresponds to the white instability region on the LHS of
Figure 5.12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.14 Bifurcation diagram for the nonlinear Mathieu equation without
delay term, equation (5.1) with α = γ = 1, β = 0. . . . . . . . . .
6.1
6.2
110
111
114
115
The phase plane analysis of the Duffing Oscillator with (a) positive
nonlinear restoring forces and (b) negative nonlinear restoring forces. 119
(a) The amplitude of the periodic motion as a function of damping.
(b) The period as function of the amplitude. . . . . . . . . . . . . 125
xii
Chapter 1
Introduction
“The goal of modeling is to find the simplest model which illustrates
the phenomenon you are interested in.”
– Richard Rand
Parametric excitation is the act of forcing a system by modulating a physical
parameter (like stiffness or moment of inertia). The mathematical result is a timevarying coefficient (usually periodic) in the governing differential equation. In
contrast with external excitation, which results in a nonhomogeneous term in the
governing differential equation, small parametric excitation can produce a large
response when the frequency of excitation is away from the primary resonance. In
this thesis, we model this phenomenon in linear and nonlinear systems where the
frequency of parametric excitation is twice that of the natural frequency of the
linear unforced system.
1
2
1.1
Motivation
Parametric excitation occurs in a wide variety of engineering applications. Here
are a few recent applications: Aubin et al  and Zalalutdinov et al  discovered
MEMS/NEMS devices are parametrically excited when illuminated within an interference field of a continuous wave laser; Stépán et al  and Kalmár-Nagy 
investigated parametric excitation in high-speed milling applications; Ramani et
al  utilized parametric excitation to model a towed mass underwater for application to submarine dynamics; Wirkus et al  observed parametric excitation
in the pumping of a swing; Yu  realized parametric excitation in a nanowire
system using an oscillating electric field; Zhang et al  noticed parametric excitation in mass-loaded string systems such as elevators, cranes and cable-stayed
bridges.
Parametric excitation dates back to Faraday in 1831  when he noticed that
surface waves in a fluid-filled cylinder under vertical excitation exhibited twice
the period of the excitation itself. Lord Rayleigh in 1883  attached a tuning
fork to the end of a stretched string and observed that when the fork vibrated
with frequency 2f , the lateral vibrations of the string responded with frequency
f . Stephenson in 1908  predicted the possibility of converting the unstable
equilibrium of an inverted pendulum by applying a vertical periodic force at the
pin. Mathieu in 1868  was investigating the vibrations of an elliptical membrane
by separation of variables when he presented the simplest differential equation that
governs the response of many systems to sinusoidal parametric excitation:
ẍ + (δ + ε cos t)x = 0
where δ and ε are constants. This equation is known as the Mathieu equation,
and has been studied extensively by many scientists. Here are a few: Bender and
3
Orzag , Minorsky , Nayfeh , Rand  and van der Pol .
In this thesis, we examine the phenomenon of parametric excitation in three different problems:
. the effect of combining 2:1 and 1:1 parametric drivers
. the effect of combining parametric excitation near a Hopf bifurcation
. the effect of delay on stabilizing a system with parametric excitation
Each of the three problems are treated separately in the following chapters of
this thesis. Each chapter is self-contained with an abstract, introduction, detailed
analysis of the problem, and conclusions. The references, however, follow the last
chapter and are listed in the order in which they appear. The main results of each
chapter are presented in the following section.
1.2
Thesis Organization
Chapter 2 is an introduction to parametric excitation demonstrated by a massspring system and the paradigm, the vertically driven pendulum. The motion of
the mass is governed by the Mathieu equation . The transition curves and
stability charts are presented here.
Chapter 3 is an examination of combining 2:1 and 1:1 two parametric drivers,
which is governed by the quasiperiodic Mathieu equation,
ẍ + (δ + ε cos t + ε cos ωt) x = 0
which we analyze in the neighborhood of the point δ = 0.25 and ω = 0.5. The
stability chart is self-similar at the resonance regions. We utilize the method of
4
multiple times scales including O(ε2 ) terms with three times scales for the small
parameter ε. By comparing our results with those of a previous work on 2:2:1
resonance , we are able to approximate scaling factors which determine the
size of the instability regions as we move from one resonance to another in the
δ-ω plane. This leads to the conjecture that the scaling of the instability tongue’s
thickness goes like ε and the curvature goes like ω.
Chapter 4 is an analysis of the effect of 2:1 parametric excitation on the stability of
a system that exhibits a Hopf bifurcation as linear damping is varied. The system
we analyze consists of a simple harmonic oscillator with small nonlinearity, small
damping and small parametric excitation in the neighborhood of 2:1 resonance:
z̈ + εAż + (1 + εk1 + εB cos 2t)z + ε(β1z 3 + β2z 2 ż + β3z ż 2 + β4ż 3 ) = 0
where A is linear damping, k1 is detuning off the 2:1 resonance, B is the amplitude
of parametric forcing, the βi ’s are the coefficients of nonlinear terms, and ε << 1.
The unforced system (B=0) exhibits the birth of a stable limit cycle as the damping changes sign from positive to negative (a supercritical Hopf bifurcation) .
We investigate the changes that occur in the steady state behavior as the linear
damping parameter is varied. A wide variety of bifurcations are observed, including
pitchforks, saddle-nodes, Hopfs, limit cycle folds, symmetry-breaking, homoclinic
and heteroclinic bifurcations. Approximate analytic expressions for bifurcation
curves are obtained using a variety of methods, including normal forms. We accompany the lowest order approximations with numerical comparisons (PPLANE 6
 and AUTO ). These give good agreement for the homoclinic and symmetrybreaking bifurcations. We show that for large positive damping, the origin is stable,
whereas for large negative damping a quasiperiodic behavior occurs. These two
5
steady states are connected by a complicated series of bifurcations which occur as
the damping is varied.
Chapter 5 is an investigation of the effect delay has on the instabilities caused
by cubic nonlinearities combined with 2:1 parametric excitation. The system we
analyze is a delayed nonlinear Mathieu equation:
ẍ + (δ + εα cos t)x + εγx3 = εβx(t − T )
in the neighborhood of δ=1/4, where δ, α, β, γ and T are parameters: δ is the
frequency-squared of the simple harmonic oscillator, a is the amplitude of the
parametric resonance, b is the amplitude of delay, c is the amplitude of the cubic
nonlinearity, and T is the delay period. The delayed linear Mathieu equation was
studied by Insperger and Stépán . The method of averaging (valid for small
ε) is used to obtain a slow flow which is analyzed for stability and bifurcations.
The bifurcations include pitchforks, saddle-nodes and a degenerate Hopf. We show
that the combined effect of 2:1 parametric excitation, cubic nonlinearity and delay
stabilizes a region in the T -δ plane for certain combinations of the delay parameters
β and T . This system would normally be unstable in the absence of delay. We also
show that the delay term behaves like effective damping, adding dissipation to a
conservative system. For example, the nonlinear Mathieu equation is known to be
a conservative system . When delay is added, the conservative system becomes
dissipative, except for certain combinations of the delay parameters β and T .
Chapter 6 offers some ideas for future work which would be an extension of the
research conducted in this thesis.
Chapter 2
Parametric Excitation
“We think our theory will provide some guidance to help engineers
avoid the problem.”
– Steven Strogatz
Parametric excitation is the act of forcing a system by modulating a physical parameter (like stiffness or moment of inertia) . The mathematical result is a
time-varying coefficient (usually periodic) in the governing differential equation.
In contrast with external excitation, which results in a nonhomogeneous term in
the differential equation, small parametric excitation can produce a large response
when the frequency of excitation is away from the primary resonance. In this
thesis, we consider linear and nonlinear systems where the frequency of parametric excitation is twice that of the natural frequency of the linear system. This
chapter examines parametric excitation by considering a mass-spring system and
a pendulum with a moving support.
6
7
2.1
Mass-Spring Oscillator
The mass-spring systems in Figure 2.1 are analyzed to demonstrate the fundamental difference between external excitation and parametric excitation. That is, a
small parametric excitation can produce a large response when the frequency of
excitation is away from the primary resonance, whereas external excitation produces a large response at the primary resonance.
The system at the top of Figure 2.1 is a mass attached to two linear springs.
The middle diagram is the same mass-spring system forced externally with F (t) =
A cos Ωt. In the bottom diagram, the second of the two springs has a time-varying
stiffness.
Let’s examine each system separately with the mass taken as one. The equation
of motion for the simple harmonic oscillator (SHO) in Figure 2.1a is:
ẍ(t) + (k1 + k2 )x(t) = 0
(2.1)
where k1 and k2 are positive. The solution to this linear constant coefficient homogeneous ODE is:
x(t) = C1 cos (ω0 t) + C2 sin (ω0 t)
where ω0 =
√
(2.2)
k1 + k2 is the natural frequency. Now, let’s consider the externally
excited SHO in Figure 2.1b. The equation of motion is:
ẍ(t) + (k1 + k2 )x(t) = A cos Ωt
(2.3)
where Ω is the forcing frequency. Notice the homogeneous part of this ODE is
equation (2.1). Hence, the complementary solution is equation (2.2). The particlar
solution is of the form:
xp(t) = C3 cos Ωt
(2.4)
8
Figure 2.1: Mass-spring oscillators: (a) simple harmonic oscillator with nat√
ural frequency of k1 + k2 , with m = 1, (b) simple harmonic oscillator with an
external forcing function, F (t) = A cos Ωt, and (c) simple harmonic oscillator
with time-varying stiffness k2 = k2 cos ωt.
However, if Ω2 = k1 +k2 , then the system is being excited at the primary resonance,
generating a secular term in the particular solution, xp (t) = C4t cos Ωt, which grows
linearly in time. However, if the system is forced away from the natural frequency,
then the particular solution is:
xp (t) = −
A cos Ωt
Ω2 − (k1 + k2 )
(2.5)
Note that as the forcing frequency approaches the linear natural frequency, Ω2 →
(k1 + k2 ), the solution, xp (t) → ∞. This is external resonance.
9
Finally, let’s consider the SHO in Figure 2.1c, where the stiffness of k2 is modulated at k2 cos ωt. The equation of motion is:
ẍ(t) + (k1 + k2 cos ωt)x(t) = 0
(2.6)
This is a linear homogeneous ODE with a periodic coefficient, where k2 cos ωt is
the parametric excitation. In general, a closed form solution does not exist for this
equation. However, we can use a perturbation method to examine the steady state
behavior of equation (2.6); this is outlined in the following section. We find that
to the lowest order approximation, a large response occurs when
√
ω 2 = 4k1 ⇒ ω = 2 k 1
(2.7)
which yields the condition that a large response occurs if the frequency of the parametric excitation is twice the natural frequency of the unforced system. This is the
motivation for examining the dynamics of systems with 2:1 parametric excitation.
Let’s examine a paradigm of parametric excitation.
2.2
Pendulum with Moving Support
A classical example of parametric excitation is the vertically forced pendulum .
Let’s consider the motion of a mass m attached to a rod of length L shown in
Figure 2.2a. The support of the pendulum is made to vibrate in the α direction by
F (t) = F cos ωt. Applying Newton’s Law of Motion in the direction perpendicular
to the rod, we obtain the equation of motion:
mL2 θ̈ = −m(g − Fy )L sin θ + mFxL cos θ
(2.8)
For small oscillations about θ = 0, with m = 1, equation (2.8) simplifies to
Fy
Fx
g
−
θ=
θ̈ +
L
L
L
(2.9)
10
Figure 2.2: The pendulum: (a) simple pendulum with a moving support,
F (t) = F cos ωt (b) inverted pendulum with moving support.
When we force the support to vibrate at F (t) = F cos ωt, Fy = F cos α cos ωt and
Fx = F sin α cos ωt. Hence,
F cos α
F sin α
g
−
cos ωt θ =
cos ωt
θ̈ +
L
L
L
(2.10)
Recall that parametric excitation appears as a variable coefficient and external excitation appears as a nonhomogeneous term. Therefore, Fy parametrically excites
the pendulum and Fx externally excites the pendulum.
Next, let’s consider the inverted pendulum whose support is made to vibrate
with F (t) shown in Figure 2.2b. The governing equation of motion is
mL2θ̈ = m(g − Fy )L sin θ + mFxL cos θ
(2.11)
11
or for small deflections about θ = 0
θ̈ + −
F cos α
g
F sin α
+
cos ωt θ =
cos ωt
L
L
L
(2.12)
The governing equations for the pendulums can be simplified by setting δ =
and ε = − FLy for the simple pendulum and δ = − Lg and ε =
Fy
L
g
L
for the inverted
pendulum, resulting in,
ẍ + (δ + ε cos t)x = Fx
(2.13)
This equation is known as the forced Mathieu equation. In the following section,
we analyze the stability of the Mathieu equation when Fx = 0.
2.3
The Mathieu Equation
The Mathieu equation,
ẍ + (δ + ε cos t)x = 0
(2.14)
is the simplest differential equation that models parametric excitation. The parameter δ is the natural frequency squared of the unforced system (ε = 0), and ε
is the amplitude of the parametric excitation. For a given pair of parameter values
(δ, ε) the system can exhibit bounded (stable) or unbounded (unstable) solutions.
We know the unforced system is stable when δ > 0 and unstable when δ < 0. It
has been shown in , , ,  that when ε 6= 0, transition curves emanate
from points on the δ-axis at
n2
δ= ,
4
n = 0, 1, 2, . . .
(2.15)
The transition curves shown in Figure 2.3 for ε < 1 separate the stable from unstable regions. Each unstable tongue is a different resonance region. The equations
12
Figure 2.3: Transition curves for the Mathieu equation for ε < 1 which
separate regions of stability (gray) from regions of instability (white).
for the transitions curves emanating from 0, 14 , 1, 94 and 4 are given out to O(ε12 )
by Rand (see Chapter 6 in ). Note that the instability region emanating from
δ = 1/4 is the largest for small ε. This region corresponds to the 2:1 parametric
excitation and results in parametric resonance.
Figure 2.3 can be used to evaluate the stability of the inverted pendulum in
Figure 2.2b when Fx = 0. As defined in the previous section, the parameter δ is
less than zero since g and L are both greater than zero. The amplitude of the
parametric excitation ε is greater than zero by similar arguments. Let’s assume
that the pendulum’s length L > F cos α. Therefore, ε < 1.
As a first example, consider the parameters (δ, ε) corresponding to the dot
labeled a in Figure 2.4. Since the parameters lie in the gray region, the equilibrium
solution for θ = 0 (standing vertical) is stable. Therefore, a small perturbation will
not cause the inverted pendulum to move to the θ = π position. Hence, vibrating
the inverted pendulum at 1 rad/sec with the small amplitude ε causes the unstable
13
Figure 2.4: Transition curves for the inverted pendulum examples. The dot
a corresponds to a stable θ = 0; the dot b corresponds to an unstable θ = 0.
equilibrium position in the unforced system to become stable.
As a second example, consider the parameters (δ, ε) corresponding to the dot
labeled b in Figure 2.4. Since the parameters lie in the white region, the equilibrium
solution for θ = 0 (standing vertical) is unstable. Therefore, a small perturbation
will cause the inverted pendulum to move to the θ = π position. In this case,
vibrating the inverted pendulum at 1 rad/sec with a small amplitude does not
change the stability. However, increasing the amplitude to any value in the gray
region would cause it to become stable.
As we have shown, parametric excitation can stabilize a system. However, it
can also destabilize a system when the parametric excitation is applied at twice
the system’s primary resonance. In the follow chapters, we examine how a system
responds to 2:1 parametric excitation in the presence of a second parametric driver
(Chapter 3), in the presence of a Hopf bifurcation (Chapter 4), and in the presence
of nonlinearities and delay (Chapter 5).
Chapter 3
2:1:1 Resonance in a Quasiperiodic
Mathieu Equation
“The important thing is not to stop questioning.”
– Albert Einstein
We present a small ε perturbation analysis of the quasiperiodic Mathieu equation
ẍ + (δ + ε cos t + ε cos ωt) x = 0
in the neighborhood of the point δ = 0.25 and ω = 0.5. We use multiple scales
including terms of O(ε2 ) with three time scales. We obtain an asymptotic expansion for the associated instability region. Comparison with numerical integration
shows good agreement for ε = 0.1. Then, we use the algebraic form of the perturbation solution to approximate scaling factors which are conjectured to determine
the size of instability regions as we move from one resonance to another in the
δ-ω parameter plane. The chapter title is explained by noting the equation above
√
may be viewed as an oscillator with natural frequency δ which is parametrically
forced with frequencies 1 and ω. Near the point δ = 0.25 and ω = 0.5, the three
frequencies are in the ratio of 2:1:1.
14
15
3.1
Motivation
The following quasiperiodic Mathieu equation,
ẍ + (δ + ε cos t + ε cos ωt) x = 0
(3.1)
has been the topic of a number of recent research papers ,-. In particular,
the stability of equation (3.1) has been investigated. Zounes and Rand  showed
that for given parameters (ε, δ, ω), equation (3.1) is said to be stable if all solutions
are bounded, and unstable if an unbounded solution exists.
3.2
Introduction
A system with two parametric drives is modeled as the quasiperiodic Mathieu
equation, represented by equation (3.1), where δ is the natural frequency squared
of the unforced system. One driver provides 2:1 parametric excitation, while the
other provides 1:1 excitation if ω = 0.5. In Chapter 2, we presented the stability
chart for the Mathieu equation, ẍ + (δ + ε cos t)x = 0. We show the lowest order
stability chart in Figure 3.1.
When a second parametric driver is added to the Mathieu equation, we retrieve
the quasiperiodic Mathieu equation (3.1). When we examine the equation to the
lowest order, we find the second driver adds a second instability tongue in the
stability chart, see Figure 3.2.
The stability charts in Figure 3.1 and 3.2 are plotted in the δ-ε plane. We
replot the stability chart in the δ-ω plane for the quasiperiodic Mathieu equation
in Figure 3.3 for ε = 0.1. Note the overlapping of the instability tongues at δ = 14 .
16
Figure 3.1: Stability chart for the Mathieu equation ε < 1. The lowest order
equation is given for the instability tongue. U is unstable, S is stable.
Figure 3.2: Stability chart for the quasiperiodic Mathieu equation ε < 1.
The lowest order equations are given for the instability tongues. U is unstable,
S is stable.
17
Figure 3.3: Stability chart for the quasiperiodic Mathieu equation in the δ-ω
plane for ε = 0.1. δ2:1 and δω:1 are the equations for the instability tongues
given in Figure 3.2. This figure is based on the lowest order analytic approximation of Figure 3.2, in contrast with Figure 3.4 which is based on the
numerical integration of equation (3.1). U is unstable, S is stable.
The stability of equation (3.1) has been investigated by Zounes . In the
previous work, it was shown for given parameters (ε, δ, ω), equation (3.1) is said to
be stable if all solutions are bounded, and unstable if an unbounded solution exists.
A stability chart in the δ-ω plane was generated for equation (3.1) with ε = 0.1,
and is shown in Figure 3.4. The chart was obtained by numerically integrating
equation (3.1). A striking feature of this complicated figure is that various details
18
Figure 3.4: Stability diagram of equation (3.1) from Zounes, . This figure
is based on the numerical integration of equation (3.1), in contrast with Figure
3.3, which is based on the lowest order analytic approximation of Figure 3.2.
White regions are unstable, black regions are stable.
appear to be repeated at different length scales (see the two rectangles). For
example, the shape of the instability regions around the point δ = 0.25 and ω = 1
appear to be similar to those near the point δ = 0.25 and ω = 0.5, except that the
latter regions are smaller in scale.
The purpose of this work is to investigate this similarity, and to obtain a scaling
law which relates the two regions boxed in Figure 3.4 above . Our approach
19
Figure 3.5: Transition curves near δ = 0.25 and ω = 1 based on equation
(3.2) from . Compare with Figure 3.4.
will be based on approximate solutions obtained by perturbation methods. In the
case of the instability regions near the point δ = 0.25 and ω = 1, a perturbation
analysis has been performed by Rand et al in , giving approximate closed form
expressions for the transition curves separating regions of stability from regions of
instability. For example, the following approximate expression was derived for the
largest of the local instability regions near δ = 0.25 and ω = 1, see Figure 3.5:
1 δ1
ω =1+ε S±
+
2
S
!!
+ O(ε2 )
(3.2)
where
δ=
1
+ δ1ε
4
and
S=
q
4δ12 − 1
(3.3)
20
In the present work we use a different perturbation method to obtain a comparable approximate expression for the largest of the local instability regions near
δ = 0.25 and ω = 0.5. By comparing the two expressions we are able to determine
scaling factors which relate their relative sizes.
3.3
Method of Multiple Time Scales
We use the method of multiple time scales on equation (3.1), with the goal of
obtaining an approximate expression for transition curves in the neighborhood of
δ = 0.25 and ω = 0.5. To accomplish this, we found it necessary to go to O(ε2 )
and to use three time scales:
ξ = t,
η = ε t,
and ζ = ε2 t
(3.4)
As usual in this method  the dependent variable x becomes a function of ξ, η,
and ζ, giving a new expression for the second derivative of x in equation (3.1):
2
2
∂ 2x
∂ 2x
d2 x
2 ∂ x
2 ∂ x
+
ε
+ O(ε3 )
=
+
2ε
+
2ε
dt2
∂ξ 2
∂ξ∂η
∂η 2
∂ξ∂ζ
(3.5)
We expand x, ω, and δ as a power series of ε:
x = x0 + ε x1 + ε2 x2 + · · ·
ω=
1
2
+ ε ω1 + ε2 ω2 + · · ·
δ=
1
4
+ ε δ1 + ε2 δ2 + · · ·
(3.6)
and substitute these series into equations (3.1) and (3.5) and collect terms in ε.
Using subscripts to represent partial differentiation, we obtain:
1
ε0 : x0,ξξ + x0 = 0
4
(3.7)
21
1
ε1 : x1,ξξ + x1 = −2x0,ξη − δ1 x0 − x0 cos ξ
4
!
ξ
− x0 cos
+ ω1 η + ω2 ζ
2
1
ε2 : x2,ξξ + x2 = −2x0,ξζ − x0,ηη − 2x1,ξη − δ2 x0 − δ1x1
4
!
ξ
+ ω1 η + ω2 ζ
− x1 cos ξ − x1 cos
2
(3.8)
(3.9)
We write the solution to equation (3.7) in the form
x0(ξ, η, ζ) = A(η, ζ) cos
ξ
ξ
+ B(η, ζ) sin
2
2
(3.10)
where A and B are as yet undetermined slowly varying coefficients. We substitute
equation (3.10) into equation (3.8) and remove resonant terms to retrieve the
following equations on A and B:
1
Aη = (δ1 − )B
2
Here, Aη =
∂A
∂η
and Bη =
∂B
.
∂η
1
Bη = −(δ1 + )A
2
(3.11)
Equations (3.11) have the solution:
A(η, ζ) = A1(ζ) cos
S
S
η + A2(ζ) sin η
2
2
The coefficient B(η, ζ) has a similar expression, where S =
(3.12)
q
4δ12 − 1. A1 and A2
are as yet undetermined slowly varying coefficients.
Having removed resonant terms from equation (3.8), we may solve for x1 , which
may be written in the abbreviated form:
x1(ξ, η, ζ) = C(η, ζ) cos
ξ
ξ
+ D(η, ζ) sin + periodic terms
2
2
(3.13)
where the first two terms on the RHS of equation (3.13) are the complementary
solution of equation (3.8) involving as yet undetermined slowly varying coefficients
22
C(η, ζ) and D(η, ζ). The “periodic terms” represent a particular solution of equation (3.8). Although the expressions for these are too long to give here, we note
that they consist of sinusoidal terms with arguments
3
S
ξ ± η,
2
2
ω1 η + ω2 ζ ±
S
η,
2
and ξ + ω1 η + ω2 ζ ±
S
η
2
(3.14)
Next we substitute the expressions for x0 and x1, equations (3.10) and (3.13), and
the expressions for A and B, equation (3.12), into the x2 equation, equation (3.9),
and eliminate resonant terms. This gives equations on C and D which may be
written in the following form:
Cη = (δ1 − 12 )D + periodic terms
(3.15)
Dη = −(δ1 + 12 )C + periodic terms
Although the expressions for the “periodic terms” again are too long to give here,
we note that they consist of sinusoidal terms with arguments
S
η,
2
(2ω1 ±
S
)η + 2ω2 ζ,
2
and
(2ω1 ±
S
)η − 2ω2 ζ
2
(3.16)
Note that equations (3.15) on the arbitrary coefficients C and D of x1 are similar in
form to the equations (3.11) on the arbitrary coefficients A and B of x0, except that
the C-D equations are nonhomogeneous. Thus our next step is to remove resonant
terms from equations (3.15). Since S =
q
in equations (3.15) which have argument
4δ12 − 1, we see that the periodic terms
S
η
2
are resonant. For general values of
ω1 , these are the only resonant terms. However, if ω1 = S/2, then some of the
other periodic terms in equation (3.15) will be resonant as well, because in that
case (2ω1 − S2 )η = S2 η.
First we consider the case in which ω1 does not equal S/2. Eliminating resonant
terms in equation (3.15) turns out to give the following equations on the slow flow
23
coefficients A1 and A2 which appeared in the expressions (3.12) for A and B:
A1ζ = −p A2
A2 ζ = p A1
(3.17)
where
p = δ1
(24δ2 − 24δ12 − 7)
12 S
(3.18)
Since all solutions to equations (3.17) are periodic and bounded, no instability is
possible.
Next we consider the resonant case in which ω1 = S/2. Elimination of resonant
terms in equation (3.15) gives:
A1ζ = −p A2 + q (−A1 sin 2ω2 ζ + A2 cos 2ω2 ζ)
A2ζ = p A1 + q ( A2 sin 2ω2 ζ + A1 cos 2ω2 ζ)
(3.19)
where p is given by equation (3.18), and where q is given by the following equation:
q=
1 δ1
+
2 S
(3.20)
As we show in the next section, equations (3.19) exhibit unbounded solutions.
Parameter values for which unbounded solutions occur in (3.19) correspond to
regions of instability in the stability diagram.
3.4
Analysis of the Slow Flow
To investigate stability, we write the slow flow system (3.19) in the form:
u̇ = −p v + q (−u sin Ωt + v cos Ωt)
(3.21)
v̇ =
(3.22)
p u + q ( v sin Ωt + u cos Ωt)
24
where A1, A2, 2ω2 and ζ have been replaced respectively by u, v, Ω and t for
convenience. We begin by transforming (3.21) to polar coordinates, u = r cos θ,
v = r sin θ:
ṙ = −q r sin(Ωt − 2θ)
(3.23)
θ̇ = p + q cos(Ωt − 2θ)
(3.24)
Next we replace θ by φ = Ωt − 2θ:
ṙ = −q r sin φ
(3.25)
φ̇ = Ω − 2p − 2q cos φ
(3.26)
Writing equations (3.25) in first order form,
dr
−q r sin φ
=
dφ
Ω − 2p − 2q cos φ
(3.27)
The general solution to equation (3.27) is:
r=√
const
Ω − 2p − 2q cos φ
(3.28)
Equation (3.28) represents a curve in the u-v plane. If the denominator of the RHS
vanishes for some value of φ, then the curve extends to infinity and the corresponding motion is unstable. This case corresponds to the existence of an equilibrium
point in the φ equation (the second of equations (3.25)). In the contrary case in
which the denominator of the RHS of equation (3.28) does not vanish, the motion
remains bounded (stable) and the φ equation has no equilibria. Thus the condition
for instability is that the following equation has a real solution:
cos φ =
Ω − 2p
2q
(3.29)
The transition case between stable and unstable is given by the condition:
Ω − 2p
= ±1
2q
⇒
Ω = 2p ± 2q
(3.30)
25
Substituting equations (3.18) and (3.20) and using Ω = 2ω2 , equation (3.30)
becomes:
(24δ2 − 24δ12 − 7)
1 δ1
ω2 = δ1
±
+
12 S
2
S
where S =
!
(3.31)
q
4δ12 − 1. In addition, for resonance we required that
ω1 =
S
2
(3.32)
Thus we obtain the following expression for transition curves near δ = 0.25 and
ω = 0.5:
1
S
(24δ2 − 24δ12 − 7)
1 δ1
ω = + ε + ε2 δ1
±
+
2
2
12 S
2
S
!!
+ ···
(3.33)
where
δ=
1
+ εδ1 + ε2δ2 + · · ·
4
(3.34)
equation (3.33) gives a value of ω for a given value of δ, the latter defined by δ1, δ2
and ε. However, for a given value of ε, any value of δ close to 0.25 can be achieved
in the form 1/4 + εδ1, that is by choosing δ2 = 0. So in our numerical evaluation
of equations (3.33) and (3.34), we take δ2 = 0. The transition curves in (3.33)
are displayed in Figure 3.6 for ε = 0.1 along with the results of previous work,
equation (3.2), already displayed in Figure 3.5. (Cf. Figure 3.4).
3.5
Discussion
We now have asymptotic approximations for corresponding instability regions at
two points in the δ-ω parameter plane, and we wish to compare them. Based on
an expansion about the point δ = 0.25 and ω = 1, we have equation (3.2) from
. And from the work presented in this paper, we have equation (3.33), valid in
the neighborhood of δ = 0.25 and ω = 0.5. We repeat these expansions here for
26
Figure 3.6: Transition curves near δ = 0.25 and ω = 0.5 based on equation
(30). Also shown for comparison are the transition curves near δ = 0.25 and
ω = 1 based on equation (3.2) from . Compare with Figures 3.4 and 3.5.
the convenience of the reader, using the subscript A for equation (3.2), and the
subscript B for equation (3.33):
1 δ1
+
ωA = 1 + ε S ±
2
S
!!
+ O(ε2 )
S
1
(−24δ12 − 7)
1 δ1
±
+
ωB = + ε + ε2 δ1
2
2
12 S
2
S
(3.35)
!!
+ O(ε3 )
(3.36)
Note that although these expansions are truncated at different orders of ε, both
are the lowest order approximations which respectively yield two transition curves
(as represented by the ± sign), and which therefore allow the thickness of the
associated instability region to be computed.
27
We compare these two expressions in two ways: 1) the centerline of the region,
and 2) the thickness of the region.
In the δ-ε plane, the centerline is the location where the tongues intersect the
δ-axis, i.e., the tongue has no thickness (cf. Figure 3.2). The respective centerlines
in the δ-ω plane are given by the following approximations:
ωA = 1 + εS + O(ε2 )
(3.37)
S
1
+ ε + O(ε2 )
2
2
(3.38)
ωB =
From these we may conjecture that the scaling of the centerline in the ω direction
goes like ω0 , being the ω value of the point of expansion. That is, in the case of
equation (3.37), ω0 = 1, while for equation (3.38), ω0 = 1/2, and we observe that
the two curves have comparable curvature, but that they are stretched in the ω
direction in proportion to their values of ω0 .
Moving on to the question of the thickness of the two instability regions, these
are obtained by subtracting the expressions for the upper and lower transition
curves, and are given by the following approximations:
thicknessA
thicknessB
1 δ1
+
= 2ε
2 S
2
= 2ε
!
1 δ1
+
2 S
+ O(ε2 )
!
+ O(ε3 )
(3.39)
(3.40)
We see that once again the two expressions have the same general form, but
that the smaller region is a factor of ε thinner than the larger region. This leads
us to the conjecture that this is due to the difference in the order of the resonance.
For example, this would lead to the guess that the comparable instability region
associated with the point δ = 0.25 and ω = 1/3 (which would correspond to a
2:
2
3
: 1 resonance) would have a similar equation for its thickness, but with an ε3
in the leading term.
28
3.6
Conclusions
We have presented a small ε perturbation analysis of the quasiperiodic Mathieu
equation (3.1) in the neighborhood of the point δ = 0.25 and ω = 0.5. We used
multiple scales and found that we needed to go to O(ε2 ) and use three time scales
in order to obtain a minimal representation of an instability region. Comparison
with numerical integration of equation (3.1) showed good agreement for ε = 0.1.
We used the perturbation approximation to estimate the scaling of instability
regions as we go from one resonance to another in the δ-ω parameter plane. This
is an interesting use of perturbation approximations. A comparable result could
not be easily achieved by purely numerical methods. For example, inspection of
Figure 3.4 would lead us to conclude that the instability region near δ = 0.25 and
ω = 1 is much thicker than the comparable region near δ = 0.25 and ω = 0.5.
However, it would be difficult to conclude using only numerical results that the
ratio of thicknesses was approximately ε, as we showed in this investigation.
Chapter 4
2:1 Resonance Near a Hopf
Bifurcation
“Never, never, never give up.”
– Winston Churchill
We investigate the dynamics of a system consisting of a simple harmonic oscillator
with small nonlinearity, damping and parametric forcing in the neighborhood of
2:1 resonance:
z̈ + εAż + (1 + εk1 + εB cos 2t)z + ε(β1z 3 + β2z 2 ż + β3z ż 2 + β4ż 3 ) = 0
where A is linear damping, k1 is detuning off the 2:1 resonance, B is the amplitude
of parametric forcing, the βi ’s are the coefficients of nonlinear terms, and ε << 1.
The unforced system (B=0) exhibits the birth of a stable limit cycle as damping
changes sign from positive to negative. Using perturbation methods and numerical
integration, we show that for large positive damping, the origin is stable, whereas
for large negative damping a quasi-periodic behavior occurs. These two steady
states are connected by a complicated series of bifurcations which occur as the
damping is varied.
29
30
4.1
Motivation
Our interest in understanding the behavior of equation (4.3) is motivated by two
applications. The first is a model of the El Niño Southern Oscillation (ENSO)
coupled tropical ocean-atmosphere weather phenomenon  and  in which
the state variables are temperature and depth of a region of the ocean called the
thermocline. The annual seasonal cycle is the parametric excitation. The model
exhibits a Hopf bifurcation in the absence of parametric excitation.
The second application involves a MEMS device  and  consisting of a
30 µm diameter silicon disk which can be made to vibrate by heating it with a
laser beam resulting in a Hopf bifurcation. The parametric excitation is provided
by making the laser beam intensity vary periodically in time.
Parametric excitation of a Hopf bifurcation was previously investigated from
a mathematical point of view , , , . In this chapter, we examine the
parametrically excited Hopf with relevance to physical applications  and .
4.2
Introduction
It is well-known that a limit cycle may be born in a process called a Hopf bifurcation. A typical setting is given by the autonomous equation:
z̈ + z + εAż + ε(β1z 3 + β2z 2 ż + β3z ż 2 + β4 ż 3) = 0
(4.1)
where ε << 1 is a perturbation parameter. Here εA is a linear damping coefficient
and the εβi are coefficients of nonlinear terms. The bifurcations associated with
the system where A = 0 was investigate by Ng and Rand -. Perturbation
theory  shows that equation (4.1) exhibits a limit cycle with amplitude r, where
r2 = −
4A
3β4 + β2
(4.2)
31
In the case that 3β4 + β2 > 0, equation (4.2) gives that the limit cycle occurs for
A < 0. From (4.1), the origin z = ż = 0 is unstable for A < 0 (negative damping),
and so the limit cycle, which has the opposite stability from that of the origin, is
in this case stable, a situation which is referred to as a supercritical Hopf.
In this chapter we investigate what happens when the system in (4.1) is parametrically excited at close to twice the natural frequency of the linearized undamped system:
z̈ + (1 + εk1 + εB cos 2t)z + εAż + ε(β1z 3 + β2z 2 ż + β3z ż 2 + β4ż 3) = 0
(4.3)
where k1 is a detuning coefficient and B is the amplitude of parametric forcing.
The linearized undamped version of equation (4.3) is known as Mathieu’s equation:
z̈ + (1 + εk1 + εB cos ωt)z = 0
(4.4)
where ω is the frequency of the parametric excitation. It is well-known  that
for small ε the largest instability occurs for ω = 2, a situation referred to as 2:1
parametric resonance, which motivates our choice of ω = 2 in equation (4.3) (cf
Chapter 2).
4.3
Two-Variable Expansion Method
We use the two-variable expansion method to investigate the dynamics of equation
(4.3) for small ε  and . We define two new time scales, ξ = t and η = εt,
where η is referred to as slow time. Equation (4.3) becomes:
zξξ + 2εzξη + (1 + εk1 + εB cos 2ξ)z + εAzξ
+ ε(β1z 3 + β2z 2 zξ + β3 zzξ 2 + β4zξ 3) = O(ε2 )
(4.5)
32
Next we expand z = z0 + εz1 + O(ε2 ) and collect terms, giving:
ε0 :
z0ξξ + z0 = 0
ε1 :
z1ξξ + z1 = −2z0ξη − k1 z0 − Bz0 cos 2ξ − Az0ξ
(4.6)
−(β1z03 + β2z02 z0ξ + β3 z0z0ξ 2 + β4z0 ξ 3)
(4.7)
We take the solution of (4.6) to be
z0 = u(η) cos ξ + v(η) sin ξ
(4.8)
where u and v depend only on slow time η. We substitute (4.8) into (4.7) and
remove resonant terms to retrieve the following slow flow:
!
A
1
k1 B
uη = − u +
−
v + (Γ1 v − Γ2 u)(u2 + v 2)
2
2
4
8
!
A
1
k1 B
+
vη = − v −
u − (Γ1 u + Γ2 v)(u2 + v 2)
2
2
4
8
(4.9)
(4.10)
Note that the coefficients β1, β2, β3, and β4 occur in the combinations 3β1 + β3
and β2 + 3β4 directly from the averaging process. We observe that this represents
a natural grouping becuase the β1 and β3 terms in equation (4.1) are conservative,
whereas the β2 and β4 are dissipative. Hence, we define:
Γ1 = 3β1 + β3,
Γ2 = β2 + 3β4
(4.11)
We may obtain an alternate form of the slow flow equations (4.9) and (4.10) by
transforming from rectangular coordinates u and v to polar coordinates r and θ
via u = r cos θ and v = r sin θ, giving:
r
rη = − (4A + Γ2 r2 + 2B sin 2θ)
8
1
θη = − (4k1 + Γ1 r2 + 2B cos 2θ)
8
where r >0 and θ depend only on slow time η.
(4.12)
(4.13)
33
4.4
Analysis of Slow Flow
In the following sections, we analyze the slow flow for equilibria and limit cycles
and examine the bifurcations that accompany any changes in stability.
4.4.1
Equilibria
Equation (4.8) tells us the nontrivial equilibria (r 6= 0) of the slow flow (4.12) and
(4.13) correspond to limit cycles in the original equation (4.3). These equilibria
satisfy rη = θη = 0:
4A + ΛΓ1 ρ + 2B sin 2θ = 0
(4.14)
4κA + Γ1 ρ + 2B cos 2θ = 0
(4.15)
where we define
ρ = r2 ,
k1 = κA,
Λ=
Γ2
β2 + 3β4
=
Γ1
3β1 + β3
(4.16)
Note that although ρ = r2 must be positive for real solutions, there are no sign
restrictions on κ and Λ.
Solving (4.14) and (4.15) respectively for sin 2θ and cos 2θ, and using the identity sin2 2θ + cos2 2θ = 1, we obtain the following condition on ρ:
(Λ2 + 1)Γ21 ρ2 + 8A (Λ + κ) Γ1 ρ + 4 4A2(1 + κ2) − B 2 = 0
(4.17)
Equation (4.17) is a quadratic on ρ. For real roots, the discriminant must be
nonnegative. This gives the condition:
1 1 + Λ2
A2
≤
B2
4 (1 − κΛ)2
(4.18)
In addition, ρ = r2 must be nonnegative. Allowable (nonnegative) values of ρ will
be separated from rejected (negative) values of ρ by the condition ρ = 0, which
34
gives:
A2
1
P = 2 =
B
4(1 + κ2 )
2
(4.19)
From this we are led to define P = A/B, the ratio of linear damping coefficient
A to parametric forcing amplitude B. P turns out to be an important bifurcation
parameter for this system. If we fix the parameters βi and κ, then we may envision
a line parameterized by P = A/B which contains 4 key bifurcation points, P1 ≤
P 2 < P3 ≤ P 4 :
v
u
P1
1 u 1 + Λ2
=− t
2 (1 − κΛ)2
1
1
P2 = − q
2 (1 + κ2 )
P3
1
1
= q
2 (1 + κ2)
1 u 1 + Λ2
P4 = t
2 (1 − κΛ)2
v
u
(4.20)
From equations (4.18) and (4.19) we see that each of these points may represent
a change in the number of slow flow equilibria. Equation (4.18) tells us that there
are no nontrivial slow flow equilibria to the left of P1 and to the right of P4 . It
also tells us that there are two real roots for ρ = r2 in the interval between P1 and
P4 . However, this does not mean that there are two allowable values of r > 0 in
this region because some or all of these roots ρ may be negative.
We now appeal to Descartes’ Rule of Signs to draw conclusions as to the number
of admissible values of r in the interval between P1 and P4 . To be specific, we
assume (Λ + κ)Γ1 and B are positive, although similar conclusions can be drawn
in the general case.
First let’s consider the case that A > 0. If P > P3 then the signs in equation
(4.17) are + + + which tells us that there are no positive roots ρ to the right of P3 ,
and hence no nontrivial slow flow equilibria to the right of P3 . If 0 < P < P3 then
the signs in equation (4.17) are + + − which tells us that there is one positive
35
and one negative root ρ, which means there is a single admissible value of r in the
interval between P = 0 and P = P3 .
Next, let’s consider the case that A < 0. If −P2 < P < 0 the signs are + − −
which again means that there is a single admissible value of r, this time in the
interval between P = P2 and P = 0. On the other hand, if P < P2 the signs are
+ − + which means that there are no negative roots ρ in the interval between P1
and P2 . However, we have seen that there are 2 real roots ρ in this interval (from
the positiveness of the discriminant), and thus we may conclude that there are 2
positive roots ρ and thus 2 admissible values for r between P1 and P2 .
Now it turns out that each of the admissible values of r corresponds to a pair
of nontrivial slow flow equilibria. This may be seen by considering the value of θ
at these equilibrium points. Eliminating ρ from equations (4.14) and (4.15) gives
− sin 2θ + Λ cos 2θ = 2
A
(1 − Λκ)
B
(4.21)
which may be written as
cos(2θ + ψ) = 2
A 1 − Λκ
√
B 1 + Λ2
(4.22)
where the phase angle ψ satisfies cot ψ = Λ. Now if A/B in equation (4.22) corresponds to a value which gives a real positive value for r, then equation (4.22)
gives two values for θ which differ by 180 degrees. Thus each admissible value of r
found above corresponds to two slow flow equilibria which are located symmetrically with respect to the origin. This may be seen by noting that equations (4.9)
and (4.10) are invariant under the transformation (u, v) 7→ (−u, −v), which means
that if (u, v) is a slow flow equilibrium, then so is (−u, −v).
In summary, we have shown that the number of slow flow equilibria (including
36
the origin r = 0) depends on the value of P = A/B, as follows:
[−∞, P1]
1 equilibrium
[P1 , P2 ]
5 equilibria
[P2 , P3 ]
3 equilibria
[P3 , +∞]
1 equilibrium
This chart is based on the following assumptions:
(Λ + κ)Γ1 > 0,
Γ2 = ΛΓ1 > 0,
B > 0,
ε << 1
(4.23)
Thus we may say that as A/B is decreased, a pitchfork bifurcation occurs as P3 is
crossed, resulting in three slow flow equilibria (the origin and the pair just born).
Continuing to decrease A/B, another pitchfork occurs as P2 is crossed, and now
there are 5 slow flow equilibria. Finally a pair of saddle-node bifurcations occur
as P1 is crossed, in which all of the four slow flow equilibria which were created
in the two pitchforks now come together in pairs and disappear. The origin is the
only remaining equilibria (see Example One).
4.4.2
Limit Cycles
The previous section summarized the occurrence of slow flow equilibria. Let’s
analyze the slow flow for the existence of limit cycles. One way that limit cycles can
occur is through a Hopf bifurcation at the slow flow equilibria. The conditions for
a Hopf can be stated in terms of the of the Jacobian matrix, which is the linearized
slow flow about an equilibrium point. Generically, for a Hopf bifurcation to occur,
the T r(J ) = 0 and Det(J ) > 0  (see Examples 2 and 3). From equations (4.9)
and (4.10), we obtain the following expression for J :
37


J =


J11 J12 
J21 J22


(4.24)
where
J11 = −
A 3
1
1
− Γ2 u2 − Γ2 v 2 + Γ1 uv
2
8
8
4
k1 B 1
3
1
− + Γ1 u2 + Γ1 v 2 − Γ2 uv
2
4
8
8
4
J12 =
J21 = −
k1 B 1
3
1
− − Γ1 v 2 − Γ1 u2 − Γ2 uv
2
4
8
8
4
J22 = −
A 3
1
1
− Γ2 v 2 − Γ2 u2 − Γ1 uv
2
8
8
4
where u and v are to be evaluated at the slow flow equilibria. From (4.24) we
obtain:
1
T r(J ) = −A − Γ2 ρ
2
(4.25)
Setting the T r(J ) = 0 for a Hopf bifurcation, we find:
1
T r(J ) = 0 ⇒ A = − Γ2 ρ
2
(4.26)
and since ρ ≥ 0, we note that for a Hopf bifurcation to occur, A must have the
opposite sign of Γ2 , which is in agreement with equation (4.2).
Next we solve (4.25) for ρ and substitute it into (4.17), obtaining the necessary
part of the Hopf condition:
2
PHopf
1
1
= 1 + 2 + 4κ κ −
Λ
Λ
−1
(4.27)
This condition, however, is not sufficient. We also need to require that Det(J ) > 0.
From equation (4.24), we obtain:
!
A2 B 2 k12
AΓ2 BΓ1 k1 Γ1
Det(J ) =
−
+
+
−
+
ρ
4
16
4
4
16
4
+
BΓ1 2 BΓ2
3 2
v −
uv
Γ1 + Γ22 ρ2 +
64
8
8
(4.28)
38
This expression may be simplified by multiplying the RHS of equation (4.9) by u
and adding it to the RHS of (4.10) multiplied by v, which gives:
uv = −
ρ(4A + Γ2 ρ)
4B
(4.29)
Similarly we multiply the RHS of (4.9) by v and subtract from it the RHS of (4.10)
multiplied by u, which gives:
v2 =
ρ(4k1 + 2B + Γ1 ρ)
4B
(4.30)
Substituting (4.29) and (4.30) into (5.55), we obtain the following expression for
Det(J ) as a function of ρ:
Det(J ) =
A2 B 2 k12 3
5 2
−
+
+ (AΓ2 + k1 Γ1 )ρ +
Γ1 + Γ22 ρ2
4
16
4
8
64
(4.31)
Recall the condition Det(J ) = 0 corresponds to a saddle-node bifurcation. Therefore, by eliminating ρ between Det(J ) = 0 in equation (4.31) and the slow flow
equilibrium condition (4.17), we retrieve the bifurcation condition:
4(κ2 + 1)A2 − B 2
4A2 κ2 Λ2 − B 2Λ2 − 8A2κΛ − B 2 + 4A2 = 0
(4.32)
Solving (4.32) for P 2 = A2/B 2 , we obtain:
1 + Λ2
A2
1
P = 2 =
,
B
4(1 + κ2 ) 4(1 − κΛ)2
2
(4.33)
which is in agreement with the values computed in equations (4.18) and (4.19).
4.5
One Parameter Bifurcations
The analytical methods thus far show that the slow flow system exhibits pitchfork,
saddle-node and hopf bifurcations. These bifurcations have one parameter normal
forms associated with them. The normal form for a pitchfork bifurcation is ẋ =
39
x(µ − x2), for a saddle-node bifurcation, ẋ = µ − x2 , and for a Hopf ṙ = r(µ1 + r2 ),
where µ is a single bifurcation parameter .
The bifurcation curves obtained in section 4.4.1 and 4.4.2 are plotted in the
A-κ parameter space in Figure 4.1 for typical parameter values of B = 1 and
Λ = −1/2 (where β1 = −6, β2 = 0, β2 = 3 and β4 = 0). A is the linear damping
and κ = k1 /A, where k1 is the detuning off of the 2:1 resonance. In keeping with
the assumptions in equation (4.23), the bifurcation set is given for κ ≤ −Λ. The
curve P1 is saddle-node bifurcation, P2 and P3 are pitchfork bifurcations, and H
is the Hopf bifurcation. These curves are given by equations (4.20) and (4.27).
Further numerical analysis reveals the slow flow system also exhibits the following bifurcations: limit cycle folds, symmetry-breaking bifurcations, heteroclinic
and homoclinic bifurcations. These are classified as two parameter bifurcations.
The numerical bifurcations curves (solid) are plotted in the Λ = −1/2 parameter
space with the analytical bifurcation curves (dotted) in Figure 4.2.
The equilibrium chart presented in section 4.4.1 shows to the right of P3 there
is one stable slow flow equilibrium and to the left of P1 there one unstable slow
flow equilibrium accompanied with a stable limit cycle. Thus, the bifurcation
set in Figure 4.2 shows that these two slow flow steady states are connected by
a complicated series of bifurcations which occur as the linear damping is varied.
Starting with positive damping and decreasing to negative values, we provide three
example with different detuning values that demonstrate the complex bifurcations
that connect those two steady states. Example One examines the slow flow for
κ = −0.8, Example Two for κ = −0.35, and Example Three for κ = 0. Each
example has the same starting and ending phase portraits, which translates to
the original system having a stable origin for positive damping and quasi-periodic
40
Figure 4.1: Analytical bifurcation set in the A-κ parameter plane for Λ =
−1/2 and B = 1. A is linear damping and κ = k1 /A, where k1 is detuning. P1
is a saddle-node, P2 and P3 are pitchforks and H is a Hopf. These bifurcation
curves are given by (4.20) and (4.27). Point Q is the location where the saddlenode and Hopf bifurcations meet tangentially.
motion for negative damping, whether we are near or away from the 2:1 resonance.
4.6
Examples
In the following section, we examine the phase portraits of three systems that
begin with positive damping and, as we decrease to negative values of damping
we see a series of complex bifurcations that connect the two stable steady states.
The choice of parameters for each example are: β1 = −2, β2 = 0, β3 = 0, β4 = 1,
B = 1, which gives Γ1 = −6, Γ2 = 3 and Λ = −1/2. Example One examines
41
Figure 4.2: Bifurcation set in the A-κ parameter plane for Λ = −1/2 and
B = 1. The solid lines are the bifurcation curves found numerically, while the
dashed were found analytically. P1 is a saddle-node, P2 and P3 are pitchforks,
H is a Hopf, LCF is a limit cycle fold, SB is a symmetry-breaking, HoC is a
homoclinic and HeC is a heteroclinic bifurcation. The numbers on the RHS,
(1), (2) and (3) reference the location of κ used in Example 1, 2 and 3.
the slow flow for κ = −0.8, Example Two for κ = −0.35, and Example Three for
κ = 0. Each example has the same starting and ending phase portraits, which
translates to the original system having a stable origin for positive damping and
quasiperiodic motion for negative damping. The only parameter to change in each
example is the linear damping, A.
42
4.6.1
Example One
We take as the first example the following system with κ = −0.80:
z̈ + (1 − ε 0.8A + ε cos 2t)z + εAż + ε(ż 3 − 2z 3 ) = 0
From equation (4.20) we obtain:
√
5 5
5
, P2 = − √ ,
P1 = −
12
2 41
5
P3 = √ ,
2 41
√
5 5
P4 =
12
(4.34)
(4.35)
and from equation (4.27) we find:
5
PHopf = √
29
(4.36)
This system has a single parameter A, and from the foregoing analysis we can say
that the number of slow flow equilibria changes with A as follows:
[−∞, −0.9317]
1 equilibrium
[−0.9317, −0.3904]
5 equilibria
[−0.3904, 0.3904]
3 equilibria
[0.3904, +∞]
1 equilibrium
From (4.36), a Hopf bifurcation may occur when A = − √529 , if Det(J ) > 0.
(Here we use the fact that A must have the opposite sign to Γ2 for a Hopf). In
order to compute Det(J ) from equation (4.31), we need ρ at the Hopf. From
(4.26) we find that ρ =
10
√
,
3 29
and substituting this value for ρ into (4.31), we find
5
Det(J ) = − 116
< 0. Thus there is no Hopf in this example.
Numerical integration of the slow flow equations (4.9) and (4.10) for the present
parameters reveals that a stable limit cycle is born in a saddle connection bifurcation at A = −0.778, not a Hopf. See Figure 4.3, bifurcation point C. Bifurcation
points A, B, and D in Figure 4.3 correspond to the points P3 , P2 , and P1 , respectively. Nothing special happens at A = − √529 = −0.9285.
43
Figure 4.3: Phase portraits of the slow flow in the (u, v) plane for Example
One with κ = −0.80. Starting with positive damping, a pitchfork bifurcation
occurs at A, followed by a second pitchfork bifurcation at B, a heteroclinic
bifurcation at C and ending with a saddle-node bifurcation at D.
44
4.6.2
Example Two
The following example exhibits qualitatively different behavior from Example One:
z̈ + (1 − ε 0.35A + ε cos 2t)z + εAż + ε(ż 3 − 2z 3 ) = 0
(4.37)
This corresponds to the choice of detuning as κ = −0.35, which lies in the region
for linear damping A < 0 which has six bifurcation curves in Figure 4.2. From
equation (4.20) we obtain:
√
10 5
10
P1 = −
,
, P2 = − √
33
449
10
P3 = √
,
449
√
10 5
P4 =
33
(4.38)
and from equation (4.27) we find:
10
PHopf = √
269
(4.39)
As with Example One, this system has the single parameter A, the linear damping
and from the foregoing analysis we can say that the number of slow flow equilibria
changes with A as follows:
[−∞, −0.6776]
1 equilibrium
[−0.6776, −0.4719]
5 equilibria
[−0.4719, 0.4719]
3 equilibria
[0.4719, +∞]
1 equilibrium
From (4.39), a Hopf bifurcation may occur when A = − √10
, if Det(J ) > 0. (Here
269
we use the fact that A must have the opposite sign to Γ2 for a Hopf). In order
to compute Det(J ) from equation (4.31), we need ρ at the Hopf. From (4.26)
we find that ρ =
Det(J ) =
40
269
√20 ,
3 269
and substituting this value for ρ into (4.31), we find
> 0. Thus, a Hopf bifurcation occurs when A = −0.6097 at the
equilibria located at ρ = 0.4065 where θ ≈ 22.7o and 202.7o from equation (4.22).
See Figure 4.4, bifurcation point F .
45
Figure 4.4: Caption for Example Two, κ = −0.35 begins on the next page.
46
Figure 4.4: Phase portraits of the slow flow in the (u, v) plane for Example
Two with κ = −0.35. Starting with positive damping, the initial flow is the same
as Example One where a pitchfork bifurcation occurs at A, followed by a second
pitchfork bifurcation at B, a limit cycle fold at C, a heteroclinic bifurcation at D,
a homoclinic bifurcation at E, a subcritical Hopf at F , ending with a saddle-node
bifurcation at G.
In order to confirm these results, and to determine any other bifurcations which
occur in (4.37) as the linear damping, A is varied, we numerically integrate the
slow flow equations (4.9) and (4.10) for the present parameters. The phase plane
analysis is displayed in Figure 4.4. We found that the limit cycles born in the Hopf
at A = −0.6097 are unstable and exist in the range −0.6097 < A < −0.5359 (from
E to F ). Prior to this bifurcation for positive damping, this system behaves like
Example 1 up to the bifurcation point B (the second pitchfork) which is followed
by the birth of a stable and unstable limit cycle in a limit cycle fold at point C,
followed by a homoclinic saddle-connection bifurcation at point D, which destroys
the unstable limit cycle. Further decreases in A results in the birth of two unstable
limit cycles in a homoclinic saddle-connection bifurcation at point E, which are
then destroyed in a subcritical Hopf bifurcation at point F. As we continue to
make A more negative, we find the pair of equilibria are destroyed in a saddlenode bifurcation at point G, leaving a stable limit cycle in the slow flow, i.e.,
quasiperiodic motion in the original system.
47
4.6.3
Example Three
As a final example, we examine the system at 2:1 resonance (κ = 0):
z̈ + (1 + ε cos 2t)z + εAż + ε(−2z 3 + ż 3) = 0
From equation (4.20) we obtain:
√
1
5
P1 = −
, P2 = − ,
4
2
1
P3 = ,
2
√
5
P4 =
4
(4.40)
(4.41)
and from equation (4.27) we find:
1
PHopf = √
5
(4.42)
As with Example One and Two, this system has the single parameter A, and from
the foregoing analysis we can say that the number of slow flow equilibria changes
with A as follows:
[−∞, −0.559]
1 equilibrium
[−0.559, −0.5]
5 equilibria
[−0.5, 0.5]
3 equilibria
[0.5, +∞]
1 equilibrium
From (4.42), a Hopf bifurcation may occur when A = − √15 , if Det(J ) > 0. (Again
we used the fact that A must have the opposite sign to Γ2 for a Hopf.) In order to
compute Det(J ) from equation (4.31), we need ρ at the Hopf. From (4.26) we find
that ρ =
2
√
,
3 5
and substituting this value for ρ into (4.31), we find Det(J ) =
3
20
> 0.
Thus, a Hopf bifurcation occurs when A = −0.4472 at ρ = 0.2981. See Figure 4.5,
bifurcation point D.
In order to confirm these results, and to determine any other bifurcations which
occur in (4.40) as A is varied, we numerically integrate the slow flow equations (4.9)
48
and (4.10) for the present parameters. The phase plane analysis is displayed in
Figure 4.5. We found that the limit cycles born in the Hopf when A = −0.447 are
unstable and exist in the range −0.447 < A < −0.377 (from C to D). As A is
increased beyond −0.377, a symmetry-breaking bifurcation occurs in which each
of the two limit cycles born in the Hopf unite into one large unstable limit cycle
which is point-symmetric about the origin in the u-v plane at point C. Further
increases in A increase the size of the unstable limit cycle until A = −0.364, at
which point it coalesces with a larger stable limit cycle in a limit cycle fold. The
behavior beyond this point for A > −0.364 is the same as Example 1 and 2. Also
note that the behavior following the subcritical Hopf is the same as Example 2.
49
Figure 4.5: Phase portraits of the slow flow in the (u, v) plane for Example
Three with κ = 0. Starting with positive damping, the initial flow is the same
as Example One where a pitchfork bifurcation occurs at A, followed by a limit
cycle fold at B, a symmetry-breaking bifurcation at C, a subcritical Hopf at
D, a second pitchfork at E, and ending with a saddle-node bifurcation at F .
The flow at  is the same in Figure 4.4.
50
4.7
Two Parameter Bifurcations
One parameter bifurcations of a system can be found analytically by linearizing
about the slow flow equilibria and analyzing the Jacobian matrix, as we did in section 4.4.2. Two parameter bifurcations cannot be found by linearization since they
are global bifurcations and usually do not occur in the neighborhood of the equilibrium . In the following sections, we describe analytical techniques to find the
symmetry-breaking bifurcation from Example 3 and the homoclinic bifurcations
of Example 2 and 3. Their normal forms are respectively :
ẋ = y,
ẏ = µ1 x + µ2 y − x3 − x2 y
(4.43)
ẋ = y,
ẏ = µ1 x + µ2 y + x2 + x2 y
(4.44)
where µ1 and µ2 is the parameter space for the bifurcations. Both normal forms
can be written as a Hamiltonian (conservative) system with a nonconservative
perturbation. This is the ansatz for the two parameter bifurcation analysis .
4.7.1
Hamiltonian System
Recall the slow flow equations for the 2:1 resonance near a Hopf bifurcation:
uη =
vη = −
Γ2
Γ1
A
κA B
−
v + (u2 + v 2)v − u − (u2 + v 2)u
2
4
8
2
8
Γ2
Γ1
A
κA B
+
u − (u2 + v 2)u − v − (u2 + v 2)v
2
4
8
2
8
(4.45)
(4.46)
Note the first two terms are conservative and can form a Hamiltonian system while
the last two terms are dissipative. Let’s momentarily remove the nonconservative
terms and treat them as a perturbation,
uη =
∂H
+ f (u, v)
∂v
(4.47)
51
vη = −
∂H
+ g(u, v)
∂u
(4.48)
where the nonconservative perturbation is:
A
Γ2
f = − u − (u2 + v 2)u
2
8
(4.49)
A
Γ2
g = − v − (u2 + v 2)u
2
8
(4.50)
The Hamiltonian system for 2:1 resonance near a Hopf bifurcation is:
∂H
=
∂v
∂H
=
∂u
Γ1
κA B
−
v + (u2 + v 2)v
2
4
8
(4.51)
Γ1
κA B
+
u + (u2 + v 2)u
2
4
8
(4.52)
We investigate the phase plane dynamics by solving the above set of equations for
the Hamiltonian, H(u, v). We begin by integrating (4.51):
κA B
H(u, v) =
−
2
4
v 2 Γ1
+
2
8
u2 v 2 v 4
+
2
4
!
+ h(u)
(4.53)
Next, we differentiate equation (4.53) with respect to u and set it equal to equation
(4.52):
Γ1
∂H
Γ1
Γ1
κA B
= (uv 2) + h0 (u) =
+
u + (uv 2) + (u3 )
∂u
8
2
4
8
8
The
Γ1
(uv 2)
8
(4.54)
term cancels out on both sides leaving an expression for h0 (u). We
integrate this term to find h(u) and the Hamiltonian:
h(u) =
κA B
+
2
4
u2 Γ 1 u4
+
2
8 4
(4.55)
The Hamiltonian for the conservative slow flow system in equations (4.51) and
(4.52) is equation (4.53) with h(u) from (4.55):
H(u, v) =
κA B
−
2
4
v2
κA B
+
+
2
2
4
2
u2 Γ 1 2
+
u + v2
2
32
(4.56)
52
Equilibria
As we mentioned, the Hamiltonian system is the conservative part of the slow flow
equations. Now, let’s look for slow flow equilibria in the Hamiltonian system:
Γ1
κA B
−
+ (u2 + v 2) v
2
4
8
Γ1 2
κA B
+
+ (u + v 2) u
2
4
8
Hv = 0 :
Hu = 0 :
(4.57)
(4.58)
First we find that the origin is always a equilibrium point. Second, from equation
(4.58) we find the first term is always nonnegative and gives nontrivial equilibria.
However, there may not be a solution to equation (4.57) if κA <
B
2
(Recall A can be
less than zero and κA can be negative or positive), which makes (u2 +v 2 ) = r2 < 0.
Thus, there are at least two nontrivial equilibria if κA <
s
u∗ = ±
In the case where κA >
B
,
2
4
3
κA B
+
2
4
B
:
2
v∗ = 0
(4.59)
the term in equation (4.57) that made r2 < 0 changes
sign to make r2 > 0. Thus, equation (4.57) yields two additional nontrivial equilibria:
∗
u =0
∗
s
v =±
4
3
κA B
−
2
4
(4.60)
and equation (4.58) yields the same two from equation (4.59).
In summary, a pitchfork bifurcation occurs when κA =
nian has two nontrivial equilibrium points for κA <
rium points for κA >
B
2
B
,
2
where the Hamilto-
and four nontrivial equilib-
B
.
2
Hamiltonian Level Curves
The phase plane for a Hamiltonian system is filled with level curves whereupon
the Hamiltonian is constant because it’s a conservative system. In the slow flow
53
Figure 4.6: Separatrix loop for the Hamiltonian System where κA <
B
.
2
system, the shape of the level curves depend on the number of equilibria in the
slow flow, and hence, the value of κA. When κA <
B
,
2
there are two centers and
one saddle point, located at the origin. For this case, there exists a trajectory
in the phase plane that connects the stable and unstable manifolds of the saddle
point. This trajectory is called a separatrix; it separates the basin of attraction
for the periodic solutions from that of the unbounded solutions. The separatrix is
also refereed to as a homoclinic orbit because the stable and unstable manifolds
that connect are from the same saddle point. See Figure 4.6.
We previously discovered that the Hamiltonian system above undergoes a pitchfork bifurcation when κA =
B
.
2
At the bifurcation point, the separatrix loop
persists, but additional equilibria are born. When κA >
B
,
2
the origin becomes a
center and as a result, two saddle points are born; there is no change to the centers.
For this case, these exists a trajectory that connects the stable manifold of one
saddle point to the unstable manifold of the other and the unstable manifold on
the former to the stable manifold of the latter. This type of separatrix is referred
to as a heteroclinic orbit. See Figure 4.7.
When nonconservative terms are added to a Hamiltonian system, the separa-
54
Figure 4.7: Separatrices for the Hamiltonian System where κA >
B
.
2
trices are destroyed. For example, a symmetry-breaking bifurcation is the splitting
of two connected homoclinic orbits into separate orbits. See Figure 4.8. The homoclinic separatrix when κA <
B
2
is the level curve of the Hamiltonian system
that passes through the origin. To find the equation of the separatrix, we need the
value of the Hamiltonian for that level curve. This is done by locating one point
on the separatrix.
Let’s change the Hamiltonian to polar coordinates for convenience. We substitute u = r cos θ and v = r sin θ into equation (4.56):
p
q
Γ2
H(r, θ) = − r2 sin2 θ + r2 cos2 θ + r4
2
2
32
where −p =
κA
2
−
B
4
and q =
κA
2
+
B
.
4
Note that for κA <
B
,
2
(4.61)
p > 0 and q > 0.
The Hamiltonian for the homoclinic orbit is found by locating one point on the
level curve. When κA <
B
2
the homoclinic orbit passes through the origin. Thus,
we know the Hamiltonian is equal to zero on the level curve. We find the equation
for the separatrix by setting H = 0 and substituting u∗ = v ∗ = 0 in equation
(4.61):
p
q
Γ2
− r2 sin2 θ + r2 cos2 θ + r4 = 0
2
2
32
(4.62)
55
Figure 4.8: Numerical homoclinic orbits for the slow flow system plotted in
the u-v plane. (a) the separatrix in the conservative system, (b) the destruction
of the separatrix when the nonconservative terms are added.
We cancel the r2 from both sides and solve for the remaining r2 , which is the radius
squared of the homoclinic orbit:
r2 =
16 q sin2 θ − p cos2 θ
Γ2
Thus, the separatrix associated with κA <
B
2
The separatrices that exists when κA >
(4.63)
is shown in Figure 4.6.
B
2
do not ass through the origin as
before because the saddle points are nontrivial. In the previous case, we knew
H = 0 on the separatrix because it passed through the origin. In this case, we
find the value of the Hamiltonian for the separatrix by substituting u∗ and v ∗ from
equation (4.60) into equation (4.61):
Γ1 + 12 κA B
−
H(u , v ) =
18
2
4
∗
∗
2
(4.64)
56
Again, we find the equation for the separatrix by setting the Hamiltonian equal to
the constant in (4.64), and solving for r2 .
Γ1 + 12 κA B
H=
−
18
2
4
2
=r
2
p
q
Γ2
− sin2 θ + cos2 θ + r2
2
2
32
(4.65)
which is rewritten as,
r
2
p 2
q
Γ2
Γ1 + 12 κA B
sin θ − cos2 θ − r2 +
−
2
2
32
18
2
4
2
=0
(4.66)
This is a fourth order polynomial on r. Thus, the separatrices are composed of
four curves. We use MACSYMA (a computer algebra program -)) to find
the roots,
r22 = −r12
(4.67)
4 √
4κA 2B cos 2θ
w−
−
r12 = − √
Γ2
Γ2
3 2Γ2
(4.68)
where w is given as:
w=
Γ22 κA2
+ (6Γ2 + 18)κA2 + (18B cos 2θ − 6BΓ2 )κA
2
BΓ22 κA 9B 2 cos 4θ B 2Γ22 3B 2 Γ2 9B 2
+
+
+
+
−
2
4
8
2
4
(4.69)
These two equations create the separatrices shown in Figure 4.7.
The separatrices derived for κA <
B
2
and κA >
B
2
in Figures 4.6 and 4.7 exists in
the Hamiltonian system only. Recall the original slow flow equations in (4.9) and
(4.10) have conservative and nonconservative terms. When the nonconservative
terms are added to the Hamiltonian system, they destroy the separatrices since
they conservative quanities. However, for particular combinations of κ and A,
for a given B and Γ2 , the separatrices may persist. Our strategy is to require
the separatrices to persist and find the associated bifurcation curve in the A-κ
parameter space for the symmetry-breaking and homoclinic bifurcations presented
in Example 2 and 3.
57
4.7.2
Preserving Closed Orbits
We wish to find the analytical bifurcation curves of the symmetry-breaking and homoclinic bifurcations presented in Example 2 and 3. The strategy is to require the
separatrices in the Hamiltonian system to persist when the perturbation (nonconservative terms) are added to the system. Let’s for a moment consider preserving
any closed trajectory in a Hamiltonian system with a small nonconservative perturbation. We call this class of systems nearly Hamiltonian. Before we investigate
this class of systems, let’s recall Green’s Theorem.
Green’s Theorem
We use Green’s Theorem to investigate the behavior of the separatrix as we vary
A and κ. Green’s Theorem gives a relationship between a line integral around any
simple closed curve C and an area integral over the plane region int C bounded by
C. We assume that int C consists of all points inside C as well as all the points on
C. The theorem states:
Let C be a positively oriented, piecewise-smooth, simple closed curve in the
plane and let int C be the region bounded by C. If P and Q have continuous
partial derivatives on an open region that contains int C, then
I
P dx + Q dy =
C
Z Z
int C
∂Q ∂P
−
∂x
∂y
!
dxdy
(4.70)
Equation (4.70) will be the basis for deriving the Melnikov Integral  for systems
that are nearly Hamiltonian, like the system in equations (4.45) and (4.46) and
looking ahead to equation (4.127).
58
Nearly Hamiltonian Systems
The autonomous system in equations (4.45) and (4.46) can be separated into a
conservative (unperturbed) and nonconservative (perturbation) part. If the conservative system has a separatrix in the phase plane, the perturbation is expected
to destroy it. Let’s define the Hamiltonian (unperturbed) system as
ẋ =
∂H
,
∂y
ẏ = −
∂H
∂x
(4.71)
where H(x, y) is the Hamiltonian function. The phase plane of Hamiltonian systems consists of trajectories with H(x, y) = constant. To see that H is constant
on any trajectory, consider the velocity of H,
dH
:
dt
∂H
∂H
dH
= ẋ
+ ẏ
dt
∂x
∂y
(4.72)
Substituting equation (4.71) into the expression above, we find
dH
=
dt
∂H
∂y
!
∂H
∂H
+ −
∂x
∂x
!
dH
dt
∂H
=0
∂y
equals zero:
(4.73)
Let’s consider a small perturbation of the Hamiltonian system in the phase plane:
ẋ =
∂H
+ εf(x, y)
∂y
(4.74)
∂H
+ εg(x, y)
∂x
(4.75)
ẏ = −
If Γε is a closed orbit in the phase plane of the perturbed system above, then
I
dH = 0
(4.76)
Γε
Actually, equation (4.76) is true for any piecewise-smooth, simple closed curve
in the plane. The Hamiltonian differential dH can also be expressed as dH =
∂H
dx
∂x
+
∂H
dy.
∂y
Therefore equation (4.76) becomes,
I
(−ẏ + εg(x, y)) dx + (ẋ − εf(x, y)) dy = 0
Γε
|
{z
∂H/∂x
}
|
{z
∂H/∂y
}
(4.77)
59
Now, let’s separate the unperturbed system from the perturbation:
I
(−ẏdx + ẋdy) + ε(g(x, y)dx − f (x, y)dy) = 0
(4.78)
Γε
Since Γε is a closed trajectory we can change the line integral to a time integral:
I
(−ẏẋ + ẋẏ) + ε(g(x, y)ẋ − f (x, y)ẏ) dt = 0
(4.79)
Γε
Clearly, the first term in the integrand vanishes. What remains is an expression
for the perturbation terms to vanish:
I
(g(x, y)ẋ − f (x, y)ẏ)dt = 0
(4.80)
Γε
This expression is an inner product between (f, g) and the normal to the trajectory;
it is this inner product whose average value is zero.
At this point, let’s change the time integral back to the line integral and apply
Green’s Theorem in the plane. This yields the Melnikov Integral :
I
g(x, y) dx − f (x, y) dy =
!
Z Z
Γε
Γε
∂g
∂f
+
dxdy = 0
∂x ∂y
(4.81)
Note equation (4.81) is an exact expression. If we think of the perturbation as
a force pushing and pulling on the closed trajectory and the integral around the
closed trajectory as an average, then, the pushing and pulling will average so to
cancel each other out.
Let’s make a further assumption about the system. If we take ε << 1, then
the perturbation is small. We can approximate the integral in (4.81) by assuming
Γε ≈ Γ, where Γ is the closed trajectory in the unperturbed system, such that
Z Z
Γ
!
∂g
∂f
+
dxdy = 0
∂x ∂y
(4.82)
This approximation can also apply to equation (4.80):
I
(g(x, y)ẋ − f (x, y)ẏ) dt = 0
Γ
(4.83)
60
These approximations to the Melnikov Integral can be used to find a relationship
between the parameters in the nonconservative perturbation so as to preserve any
closed loop in the phase plane, .
4.8
Symmetry Breaking Bifurcation
Let’s return to the slow flow equations (4.45) and (4.46) and assume κA <
B
.
2
Re-
call the equilibria in the system are two centers and one saddle point. The condition to preserve the separatrix is simply the area integral of the normal component
of the nonconservative perturbation about the unperturbed separatrix, equation
(4.82). To compute this integral, we need the derivatives of equations (4.49) and
(4.50):
∂f
A Γ2
= − − (3u2 + v 2)
∂u
2
8
∂g
A Γ2
= − − (u2 + 3v 2)
∂v
2
8
(4.84)
(4.85)
The integrand of the separatrix condition is the addition of (4.84) and (4.85)
∂f
∂g
Γ2
+
= −A − (u2 + v 2)
∂u ∂v
2
(4.86)
Since we found the separatrix in polar coordinates, we transform the integral and
the integrand to polar coordinates:
Z
θ0
Z
0
where r0 =
16
Γ2
r0
Γ2
A + r2 r drdθ = 0
2
0
(4.87)
q sin2 θ − p cos2 θ . Recall this equation for r comes from equation
(4.63). Let’s continue by integrating with respect to r:
Z
θ0
"
0
Z
θ0
Ar2 Γ2 r4
+
2
8
2
#r0
dθ =
0
2
2
2
A q sin θ − p cos θ + 16 q sin θ − p cos θ
0
2 dθ = 0
(4.88)
61
The integral in equation (4.88) is not difficult to compute, but it is messy. We use
MACSYMA to solve this integral. The solution yields a condition on A and κA,
for a given B (notice Γ2 is not part of the condition):
√
2
B − 2κA 32κA − 8AκA + 4B
√
2
√
!
B + 2κA
arctan √
+
B − 2κA
B + 2κA −24κA2 + (12B + 4A)κA − 2AB = 0
(4.89)
The parameter κA cannot be solved for analytically from the expression above.
Thus, we employ a root-by-bisection program in MACSYMA to find the values of
κA for a given A and B. The result is a curve in the A-κ parameter plane that
represents when the separatrix loop persists, which only happens at the bifurcation
point.
As in the previous examples, we set B = 1, Γ1 = 6, and Γ2 = −3. The analytical
curve for the bifurcation set of Λ = − 12 is plotted with the numerical curve in Figure
4.9. Notice that the first order approximation does not completely fit the numerical
curve. One explanation stems from the nature of the approximation. When we
used equation (4.82), we choose to integrate about the unperturbed separatrix.
We did not account for the changes in the shape of the separatrix due to the
perturbation. Also, we took the slow flow as nearly Hamiltonian by assuming the
nonconservative perturbation was of order epsilon. Actually, the conservative and
nonconservative parts of the slow flow areboth of order epsilon as a results of the
two-variable expansion.
62
Figure 4.9: Bifurcation set in the A-κ parameter plane for Λ = −1/2 and
B = 1. P1 is a saddle-node, P2 and P3 are pitchforks, H is a Hopf, LCF is
a limit cycle fold, HoC is a homoclinic and HeC is a heteroclinic bifurcation.
SBn is a symmetry-breaking bifurcation curve found by numerical integration
and SBa is the symmetry-breaking bifurcation curve found analytically.
63
4.9
Homoclinic Bifurcation
Recall that the slow flow system for 2:1 resonance near a Hopf bifurcation is:
u0 =
Γ2
Γ1
A
κA B
−
v + (u2 + v 2)v − u − (u2 + v 2)u
2
4
8
2
8
Γ1
A
Γ2
κA B
u − (u2 + v 2)u − v − (u2 + v 2)v
v =−
+
2
4
8
2
8
0
(4.90)
(4.91)
where the prime denotes differentiation with respect to slow time, η. For the
examples completed in the previous sections, we chose Λ = Γ2 /Γ1 = −3/6 and
B = 1. In section 4.4.2 we obtained analytical expressions for the Hopf bifurcation,
the pitchfork bifurcation, and the saddle-node bifurcation. In Figure 4.1, we label
point Q as the location where the Hopf bifurcation curve meets the saddle-node
bifurcation curve. As a result, this point is singular.
Limit cycles can be generically created or destroyed in a Hopf bifurcation. The
necessary conditions for a generic Hopf bifurcation in a system of two first order
ODE’s such as the slow flow (4.90) and (4.91) are T r(J ) = 0 and Det(J ) > 0,
where J is the Jacobian matrix evaluated at the equilibria. The condition for a
saddle-node bifurcation is the Det(J ) = 0. Thus, when these two bifurcations
occur simultaneously, as at point Q, the T r(J ) = 0 and Det(J ) = 0. When this
happens, the Jacobian matrix has a double zero eigenvalue. The Jacobian matrix
associated with the slow flow vector field in (4.90) and (4.91) evaluated at the
equilibria where these bifurcations occur and with the value of κ and A associated
with point Q is:


J =

0
0

1 
0


(4.92)
The goal of this section is to study the dynamics in the neighborhood of the
equilibria having the linear part above. This is accomplished by embedding the
64
system in a parameterized family of systems where the number of parameters
is equal to the codimension of the bifurcation . The parameterized family
of systems are called unfoldings. Therefore, we transform the slow flow system
into the following two-parameter family of systems, which is the normal form for
unfolding the singularity of a matrix with two zero eigenvalues:
ẏ = λ1 + λ2 y + x2 + xy
ẋ = y,
(4.93)
where the λ’s are the bifurcation parameters . This normal form is widely
known as the Takens-Bogdanov normal form  and . In our work, we rename
the coefficients in equation (4.93),
ẏ = µ1 + µ2 y + µ3 x2 + µ4 xy
ẋ = y,
(4.94)
where µ1 and µ2 are the linear coefficients and µ3 and µ4 are the nonlinear coefficients. After we transform the slow flow system into the normal form above, the
µ’s will be functions of the two bifurcation parameters κ and A. In the following
subsection, we outline the necessary steps taken to complete this transformation.
4.9.1
Double Zero Eigenvalue
The investigation of the dynamics of any system begins with finding equilibria
and evaluating their stability. In this case, we also are interested in finding the
non-saddle slow flow equilibria and the equilibria where the trace and determinant
vanish. Thus, we evaluate the Jacobian matrix of the slow flow in (4.90) and (4.91):


JQ = 

A
2
−
9 2
u
8
−
− κA
2
1
4
−
3 2
v
8
−
3
uv
2
− 34 + 34 v 2 + 94 u2 − 34 uv
κA
2
−
1
4
3 2
u
4
−
A
2
− 98 v 2 − 38 u2 + 34 uv
−
9 2
v
4
−
3
uv
4




(4.95)
65
The trace and determinant of the Jacobian is:
3
T r(JQ ) = A − (u2 + v 2)
2
Det(JQ ) =
(4.96)
1 2
3
A (1 + k 2) + A(1 − 2κ)(u2 + v 2 )
4
4
+
3
1
135 2
(u + v 2)2 − (v 2 + uv − v 2) −
64
8
16
(4.97)
When we solve the four equations, uη = 0, vη = 0, T r(JQ ) = 0 and Det(JQ ) =
0 simultaneously, we retrieve the (u, v) coordinates of the nontrivial slow flow
equilibria and the location of point Q in the A-κ parameter space:
s
s
q
2 √
u ≡ u1 = ±
5+1
15
2
A ≡ A1 = − √
5
q
2 √
v ≡ v1 = ±
5−1
15
(4.98)
3
4
(4.99)
κ ≡ κ1 = −
Point Q is − √25 , − 34 , which we call (A1 , κ1) in the A-κ parameter space. Let’s
linearize about the location of the singularity:
3
κ = κ1 + εκ2 = − + εκ2
4
(4.100)
2
A = A1 + εA2 = − √ + εA2
5
(4.101)
and linearize about the location of the slow flow equilibria:
u = u1 + εu2 =
v = v1 + εv2 =
s
s
q
2 √
5 + 1 + εu2
15
(4.102)
q
2 √
5 − 1 + εv2
15
(4.103)
The purpose of unfolding the singularity is to capture the dynamics in the
neighborhood of the degenerate equilibria. Therefore, by making the transformations in (4.102) and (4.103), we move the coordinate system to the degenerate
equilibria, as shown in Figure 4.10.
66
√
Figure 4.10: Phase portrait of the slow flow at κ = −3/4 and A = −2/ 5.
The first transformation is moving the system from the origin to the location
of the singular slow flow equilibria at (u1, v1).
We substitute the transformations (4.100), (4.101), (4.102) and (4.103) into the
slow flow equations (4.90) and (4.91), solve for u02 and v20 and taylor expand the
equations out to O(ε):
u02 = b1A2 + b2 κ2 + b3u2 + b4v2 + ε ( quadratic terms )
(4.104)
v20 = b5A2 + b6κ2 + b7u2 + b8v2 + ε ( quadratic terms )
(4.105)
where the “quadratic terms” are linear combinations of A2u2, A2 v2, κ2 u2, κ2 v2, u22,
v22 u2 v2, and the bi ’s are the coefficients of the linear terms:
√
30
s1 −
b1 = −
40
√
30
b5 = −
s1 +
30
where s1 =
q√
√
√
√
1
30
6
5+1
s2 , b2 = −
s1, b3 = − , b4 = −
30
15
2
4
√
√
√
1
30
6
5−1
s2, b6 =
s2 , b7 =
, b8 =
(4.106)
40
15
4
2
5 − 1 and s2 =
q√
5 + 1. Therefore, s1s2 = 2.
67
Continuing, we diagonalize the linear part of the (u2, v2) equations, which results in a new coordinate system (u3, v3) whose axes lie in the eigendirections of
the degenerate equilibrium point. Thus, we generate a new transformation:
u2 = B1u3 + B2 v3
such that
u03 = v3 + N.L. + Unfolding
v30
v2 = B3 u3 + B4 v3
(4.107)
= 0 + N.L. + Unfolding
where the Bi ’s are to be found, “N.L.” refers to the quadratic nonlinear terms and
“Unfolding” refers to A2 -κ2 unfolding terms. When the “ N.L.” and “Unfolding”
are omitted, equations (4.104) and (4.105) become,


u0 = bu ⇒ b = 

b3



=

b4 

b7
b8


√
−
1
2
5+1
4
−



√
5−1
4
1
2
(4.108)
In order to solve u02 = b3 u2 + b4 v2 and v20 = b7 u2 + b8v2 , we diagonalize the b-matrix
in (4.108) by creating the rotation matrix B such that




u03
v30






= [B]−1[ b ][B] 





=

u3 
v3

0
0






1   u3 
0
v3
(4.109)
The rotation matrix B yields the following transformation from (4.107):
u2 = v3
v2 = (1 −
√
√
5)/2 u3 + (1 − 5) v3
(4.110)
We substitute the transformation (4.110) into the slow flow equations (4.104) and
(4.105), solve for u03 and v30 and taylor expand the equations out to O(ε):
u03 = v3 + c1 A2 + c2κ2 + ε ( quadratic terms )
(4.111)
v30 = 0 + c3A2 + c4 κ2 + ε ( quadratic terms )
(4.112)
68
where the quadratic terms are linear combinations of A2u3, A2 v3, κ2 u3, κ2 v3, u23,
v32, u3v3 , and the ci ’s are the coefficients of the linear terms:
c1
c3
√
√
√
30
30
6
s1 −
s2 ,
s1
= −
c2 = −
40
30
15
√
√
√
√
√
√
√
2 6 + 30
30 − 3 6
6
30 − 6
s1 +
s2 , c4 =
s1 −
s2 (4.113)
=
48
96
30
60
where s1 =
q√
5 − 1 and s2 =
q√
5 + 1.
Using a near-identity transformation, we convert the (u3, v3) slow flow equations
into the normal form equations (4.94). The form of the near-identity transformation is:
u3 = u4 + ε(C1u24 + C2u4 v4 + C3v42
+ C4 A2u4 + C5A2 v4 + C6 κ2 u4 + C7κ2 v4)
v3 = v4 + ε(C8u24 + C9 u4v4 + C10v42
+ C11A2u4 + C12A2v4 + C13κ2 u4 + C14 κ2 v4)
(4.114)
where the Ci ’s are to be chosen. We substitute the transformation (4.114) into the
slow flow equations (4.111) and (4.112), solve for u04 and v40 and taylor expand the
equations out to O(ε):
u04 = v4 + d1 + d2 A2u4 + d3 A2v4 + d4 κ2u4
+ d5 κ2 v4 + d6 u24 + d7 u4 v4 + d8 v42
(4.115)
v40 = 0 + d9 + d10A2 u4 + d11 A2v4 + d12 κ2 u4
+ d13κ2 v4 + d14u24 + d15 u4v4 + d16 v42
(4.116)
69
where the di ’s are constants that depend on the Ci ’s in equation (4.114), omitted
for brevity. The task here is to simplify equations (4.115) and (4.116) by choosing
the constants Ci ’s judiciously. We choose to eliminate the following coefficients:
d2 , d3 , d4 , d5 , d6 , d7 , d8 and d16, giving the simplified system:
u04 = v4 + d1
(4.117)
v40 = 0 + d9 + d10 A2u4 + d11A2 v4 + d12 κ2u4 + d13 κ2v4 + d14 u24 + d15 u4v4 (4.118)
This leads to values for the Ci ’s, which are not listed here, again for brevity. Now
that we have the desired Ci ’s, and hence di ’s, we convert the two first order slow
flow equations in (4.117) and (4.118) to a second order differential equation on u4:
(
d2 u4
du4
du4
2
+
D
=
D
+
ε
D
+
D
u
+
D
u
+
D
u
1
2
3
4
4
5
6
4
4
dη 2
dη
dη
)
(4.119)
where the Di ’s are:
D1 =
D2 =
D3 =
D4 =
D5 =
D6 =
where s1 =
q√
√
√
√
√
5 6 + 30)
6 + 30
s1A2 −
s1κ2
96
60
√
√
6+5 6
s1 A2κ2
−
200
√
√
1−2 5
5+6
A2 +
κ2
−
16
10
√
6
s1
−
4
√
8 5 − 16
κ2
25
√
30
s1
−
5
5 − 1 and s2 =
q√
5 + 1.
(4.120)
70
As stated at the end of the previous section, our goal is to put equation (4.119)
into the normal form:
ẋ = y,
ẏ = µ1 + µ2 y + µ3 x2 + µ4 xy
(4.121)
which can be written as a second order ODE:
ẍ = µ1 + µ2 ẋ + µ3 x2 + µ4 xẋ
(4.122)
Note that this equation has no linear x-term, whereas equation (4.119) contains a
linear u4-term. Thus, we eliminate the u4-term by generating a final transformation: u5 = u4 +
D3
.
2D4
This yields the final slow flow equation:
du5
du5
d2 u5
+ F 4 u5
= F1 + F2 u25 + F3
2
dη
dη
dη
(4.123)
where the Fi ’s are:
√
√
5A2
2κ2
13 30 + 15 6
√
εA2κ2
− √
−
F1 = √
4 30s1 5 6s1 4800 5 − 14400
√
√
√
√
25 30 − 41 6
29 30 + 19 6
2
√
√
−
εs1 κ2 −
εs1 A22
1200 5 − 3600
3072 5 − 9216
√
6
εs1
F2 = −
(4.124)
4
√
√
6 5+5
7 5−6
F3 =
εA2 +
εκ2
40
25
√
√
3 30 − 5 6
√
εs1
F4 =
5 5 − 15
q√
q√
where s1 =
5 − 1 and s2 =
5 + 1. The first order form of the (u5, v5) slow
flow is:
u˙5 = v5 ,
v˙5 = F1 + F3 v5 + F2u25 + F4u5 v5
(4.125)
where it is exactly the Takens-Bogdanov normal form when we set µ1 = F1,
µ2 = F3, µ3 = F2 and µ4 = F4.
71
Let’s recap the transformation with a “road map”:
(u, v)
original slow flow coordinates for 2:1 resonance near a Hopf
(u1 , v1)
location of the degenerate equilibrium point in the (u, v) coordinates
(u2 , v2)
translation of (u, v) coordinates such that the origin in (u2 , v2)
coordinates is at the degenerate equilibrium point
(u3 , v3)
rotation of (u2, v2) coordinates such that the axes of (u3, v3 )
system lie along the eigendirections of degenerate equilibrium point
(u4 , v4)
near identity transformation to simplify the form of the equations
(u5 , v5)
linear transformation puts equations in Takens-Bogdanov
normal form
4.9.2
Homoclinic Bifurcation Revisited
The slow flow equations for the 2:1 resonance near a hopf bifurcation were converted to the Takens-Bogdanov normal form by a series of five transformations
outlined above. For simplicity, let’s use the normal form in equation (4.121). As
outlined in section 4.7.1, this system of equations can be written as a sum of a
conservative and a nonconservative part :
ẋ = y
ẏ =
µ1 + µ3 x2 + µ2 y + µ4 xy
|
{z
}
Conservative
|
{z
}
(4.126)
Nonconservative
Thus, we can express the system as function of a Hamiltonian and a nonconservative terms:
ẋ =
∂H
∂y
ẏ = −
∂H
+ g(x, y)
∂x
(4.127)
72
Figure 4.11: Phase portrait of the Takens-Bogdanov normal form.
where g(x, y) = µ2 y + µ4 xy. The Hamiltonian for this system when g(x, y) = 0 is:
H(x, y) =
y2
x3
− µ1 x − µ3 = constant
2
3
(4.128)
Recall that the second transformation, (u1 , v1) 7→ (u2 , v2), moved the coordinate
system from the origin to the degenerate equilibrium point. Analyzing the TakensBogdanov normal form reveals that in neighborhood of the degenerate point, there
are two equilibrium points:
(x∗1, y1∗) =
q
−µ1 /µ3 , 0 ,
q
(x∗2, y2∗) = − −µ1 /µ3 , 0
(4.129)
Conversely, when µ1 /µ3 > 0, there are no equilibrium points and when µ1 = 0,
there is one equilibrium (the degenerate case). Let’s investigate this further.
When we evaluate the Jacobian matrix of the conservative part of the slow flow
in (4.126) at each of the equilibrium points in (4.129), we find the T r(JQ ) = 0 at
both. However, the Det(JQ ) > 0 for one and Det(JQ) < 0 for the other. Therefore,
the latter is a saddle point and the former is a center. Since µ1 = F1 and µ3 = F2
are complicated functions of A2 , κ2 and ε, we assume the Det(x∗1 , y1∗) > 0 and
Det(x∗2, y2∗ ) < 0, whereupon the phase portrait for the Hamiltonian system is
shown in Figure 4.11.
73
The trajectory connecting the stable and unstable manifolds of the saddle point
is called the separatrix, which separates the basin of attraction for the periodic
solutions from that of the unbounded solutions. The periodic solutions lie inside
the region of the separatrix on the right.
To proceed as wed did in section 4.8, we would like to know the function that
defines the separatrix. The first step is to find the value of H on the separatrix.
Thus, we evaluate the Hamiltonian at the equilibrium point: (x∗2 , y2∗), which yields
the H = constant:
q
H − −µ1 /µ3 , 0 = µ1
s
q
µ3
2µ1
µ1
−µ1 /µ3 + (−µ1 /µ3 )3/2 =
− =h
3
3
µ3
(4.130)
where h is the constant on the separatrix and µ1 = F1 and µ3 = F2 in (4.124).
However, this separatrix does not exist when there is one or zero equilibrium
points in the slow flow. Remember, the analysis thus far is for the conservative
system. We learned that when the nonconservative perturbation is added to the
Hamiltonian system, the separatrix disappears. Therefore, we need to investigate
how the separatrix changes when we vary the nonconservative parameters µ2 and
µ4 , which are functions of A and κ. Hence, the unfolding has uncovered the
homoclinic bifurcation. To continue, let’s return the nearly Hamiltonian system:
ẋ =
∂H
∂y
ẏ = −
∂H
+ g(x, y)
∂x
(4.131)
where g(x, y) = µ2 y + µ4 xy and µ2 and µ4 are order of epsilon. Using the approximation to the Melnikov Integral in equation (4.83) with f = 0, the condition for
the preservation of the separatrix loop (to find the bifurcation curve in the A-κ
parameter space) is:
I
Γ
(µ2 y + µ4 xy)ẋ dt =
I
Γ
(µ2 + µ4 x)ẋ2 dt = 0
(4.132)
74
where x(t) is the separatrix. In equation (4.130), we found
s
2µ1
µ1
ẋ2
x3
h=
− µ1 x − µ3
− =
3
µ3
2
3
(4.133)
Equation (4.128) can be rewritten as a nonlinear first order ODE on x, when we
define ẋ = y:
1
2
dx
dt
!2
x3
= µ1 x + µ3 + h
3
(4.134)
We solve for x(t) above by using separation of variables:
Z
x3
µ1 x + µ3 + h
3
!−1/2
dx =
Z
√
2dt
(4.135)
Fortunately, the integrand factors:
s
µ1
x3 2µ1
µ3
µ1 x + µ3 +
− =
3
3
µ3
3
Let’s define η1 =
x+
s
µ1
−
µ3
!2
s
µ1
x−2 −
µ3
!
(4.136)
q
−µ1 /µ3 and η2 = µ3 /3 for convenience. Thus, the integral in
equation (4.135) simplifies to:
Z
(x + η1 )
−1
(x − 2η1 )
−1/2
dx =
Z q
(4.137)
q
(4.138)
2η2 dt
Again we use MACSYMA to solve the integral,
2
x − 2η1
√
arctan
3η1
3η1
provided η1 =
!1/2
=
2η2 t
q
−µ1 /µ3 > 0, which is true if µ3 < 0. Hence,
x(t) = 3η1 tan
2
q
6η1 η2t/2 + 2η1
(4.139)
provided η2 > 0. However, η2 = µ3 /3 < 0 (cf. equation (4.124)). Therefore,
√
η2
is imaginary. This works to our benefit since tan(ix) = i tanh(x). Finally,
x(t) = −3η1 tanh
2
q
6η1 η2 t/2 + 2η1
(4.140)
75
Let’s return to equation (4.132), the condition on A and κ to preserve the separatrix,
I
Γ
(µ2 + µ4 x)ẋ2 dt = 0
(4.141)
Substituting equation (4.140) in the integral above, and computing the integral in
MACSYMA, we retrieve:
√
√
8 5
2336 5 2
A2 =
κ2 +
κ
25
3125 2
(4.142)
We want this bifurcation curve in the original parameter space of A-κ. Therefore, we substitute equation (4.100) and (4.101) into equation (4.142) to yield the
homoclinic bifurcation curve:
√ √ 8 5
3
2336 5
3 2
2
A=
κ+
+
κ+
−√
25
4
3125
4
5
(4.143)
Figure 4.12 shows the first order analytical curve in the A-κ parameter space. For
parameter values in the neighborhood the singular point − √25 , − 34 , the analytical
curve HoCa is in very good agreement with the numerical curve HoCn .
4.10
Discussion
The main difference between the three examples is that Examples 2 and 3 involve
the occurrence of a Hopf bifurcation, whereas Example 1 does not. Since each
examples correspond to the parameter value Λ = −1/2, we may gain insight into
the dependence of the dynamical structure on parameters by varying κ and A as in
Figure 4.2. Here Example 1 corresponds to the horizontal line κ = −0.80, Example
2 corresponds to the horizontal line κ = −0.35 and Example 3 corresponds to
the horizontal line κ = 0. As can be seen from this figure, the branch of Hopf
bifurcations exists for κ > −3/4. As expected, the horizontal line κ = −3/4, which
76
Figure 4.12: Bifurcation set for Λ = −1/2. Point Q is the location where
the double zero eigenvalue occurs. P1 is the saddle-node bifurcation, H is the
Hopf bifurcation, HoCn is the homoclinic bifurcation curve found by numerical
integration and HoCa is the homoclinic curve found by unfolding point Q.
corresponds to a system for which both the T r(J ) and the Det(J ) simultaneously
vanish, lies between Examples 1 and 2.
In addition to the saddle-node, pitchfork and Hopfs bifurcations, which are
shown in Figure 4.1, and for which we obtained analytical expressions. This system
also exhibits limit cycle folds, symmetry-breaking, homoclinic and heteroclinic
bifurcations. We have found analytical expressions for the symmetry-breaking
and homoclinic bifurcations. We obtained numerical approximations for these
bifurcations and they are shown in Figures 4.9 and 4.12.
Although Figures 4.3-4.5 are drawn for the specific value Λ = −1/2, certain
77
features of the bifurcation set will occur for a generic value of Λ. These include:
1. Places where the Hopf curve becomes tangent to the P1 or P4 saddle-node
bifurcation curve and terminates. As just mentioned, both the T r(J ) and the
Det(J ) vanish at such a point. The corresponding value of κ is κ =
1
2Λ
− Λ2 .
2. Places where the P1 and P2 bifurcation curves are tangent. By symmetry, the
P4 and P3 curves are also tangent there. The associated value of κ is κ = −Λ.
3. Places where the P1 and P4 branches go off to infinity. These come from the
vanishing of the denominators in equation (4.20) and correspond to κ = Λ1 .
4.11
Conclusions
We have studied the dynamics of a system (4.3) which exhibits the simultaneous
phenomena of both parametric excitation and Hopf bifurcation. Imagine that we
hold all parameters fixed except for the damping coefficient A, and that we ask
what is expected to happen as A decreases through the parametric resonance/Hopf
region. We begin with a stable equilibrium point at the origin of the slow flow which
corresponds to a stable trivial solution of equation (4.3). Then the first bifurcation
is reached while A is still positive, giving rise to a pair of stable equilibria in
the slow flow which correspond to a single stable period-2 subharmonic motion
in equation (4.3). As A decreases and becomes negative, a variety of bifurcations
may occur which culminate for sufficiently negative A in a slow flow which exhibits
only a stable limit cycle and an unstable equilibrium at the origin. This limit
cycle corresponds in equation (4.3) to a stable quasiperiodic motion which may
be thought of as combining the periodic motion which would have been created in
the Hopf (in the absence of parametric excitation) with a motion coming from the
78
periodic parametric forcing.
In many applications, a limit cycle created in a Hopf bifurcation is destroyed in
a saddle-connection bifurcation in the unforced system. This scenario occurs, for
example, in the Takens-Bogdanov (double-zero eigenvalue) system (see , p.371).
Such a situation also occurs in the ENSO system - described briefly in the
introduction. When such a system is parametrically excited, the analysis in the
present work shows that a quasiperiodic motion may be expected to occur in the
neighborhood of the Hopf bifurcation. We begin to examine what happens to that
motion as the bifurcation parameter proceeds towards the value corresponding to
the saddle-connection bifurcation in the unforced system Chapter 6.
Chapter 5
2:1 Resonance of a Delayed
Nonlinear Mathieu Equation
“The lecturer should give the audience full reason to believe that all
his powers have been exerted for their pleasure and instruction.”
– Michael Faraday
We investigate the dynamics of a delayed nonlinear Mathieu equation:
ẍ + (δ + εα cos t)x + εγx3 = εβx(t − T )
in the neighborhood of δ=1/4. Three different phenomenons are combined in this
system: 2:1 parametric resonance, cubic nonlinearity, and delay. The method of
averaging (valid for small ε) is used to obtain a slow flow which is analyzed for
stability and bifurcations. We show that the combined effect of 2:1 parametric
excitation, cubic nonlinearity and delay stabilizes a region in the T -δ plane for
certain combinations of the delay parameters β and T , which would normally be
unstable in the absence of delay. We also show that the delay term behaves like
effective damping, adding dissipation to a conservative system.
79
80
5.1
Motivation
Insperger and Stépán generated the stability chart for the delayed linear Mathieu
equation with a delay period of T = 2π . Our analysis shows that the behavior
of the system for T = 2π is a very special case (especially relevant for milling
applications, ) in that it is qualitatively different from that of systems for T
is different from T = 2π. We present an investigation of the delayed nonlinear
Mathieu equation with varying delay period, valid for small ε and small εT .
5.2
Introduction
This chapter concerns the dynamics of the following parametrically excited, nonlinear differential delay equation (DDE):
ẍ(t) + (δ + a cos t) x(t) + cx(t)3 = bx(t − T )
(5.1)
where δ, a, b, c and T are parameters: δ is the frequency-squared of the simple
harmonic oscillator, a is the amplitude of the parametric resonance, b is the amplitude of delay, c is the amplitude of the cubic nonlinearity, and T is the delay
period.
Various special cases of equation (5.1) have been studied previously, depending
on which parameters are zero. In the case that b and c are zero, we have the linear
Mathieu equation, the stability chart for which is well known (Cf. Chapter 2), see
Figure 5.1. In the case that only b is zero, we have a nonlinear Mathieu equation
for which the bifurcations associated with stability change are also well known .
In the case that a and c are zero, we have the linear autonomous DDE of Hsu and
Bhatt , who generated stability charts, see Figure 5.2.
81
Figure 5.1: Stability chart for the Mathieu equation without delay, equation
(5.1) with b = c = 0. The solution is stable in the shaded regions.
Figure 5.2: Stability chart for the Hsu-Bhatt DDE, equation (5.1) with a =
c = 0 and T = 2π. The solution is stable in the shaded regions.
82
Figure 5.3: Stability chart for the delayed Mathieu equation with T = 2π,
a = 1 and c = 0, . The solutions in the shaded regions are stable. The darker
shaded triangles are those in Figure 5.2.
The linear form of equation (5.1) has been studied previously by Insperger and
Stépán , who, by utilizing the method of exponential multipliers, generated the
stability chart for a fixed delay period T =2π, where a=1 and c=0, see Figure 5.3.
In this research effort we use the method of averaging to generate stability
charts and associated bifurcations for the delayed nonlinear Mathieu equation (5.1)
with a general delay period . We note that although DDE’s are infinite dimensional , and hence more complicated than ODE’s, the averaging method which
we use replaces the original DDE by an ODE slow flow, an approximation which
is valid for small ε.
83
5.2.1
The Hsu-Bhatt Equation
The following simple harmonic oscillator with a delay term is the Hsu-Bhatt equation,
ẍ(t) + δx(t) = bx(t − T )
(5.2)
where b is the delay amplitude, the T is the delay period, and δ is the frequencysquared of the simple harmonic oscillator. If the delay amplitude was zero (b=0),
then the simple harmonic oscillator is stable when δ > 0 and unstable when δ < 0.
When we turn up (or down) the value of the delay amplitude and fix the system a
delay period, the stable system can become unstable and the unstable system can
become stable.
Let’s consider the solution to equation (5.2). Generally, to solve a second order
constant coefficient differential equation, we assume the solution to be of the form:
x(t) = eλt
(5.3)
Substituting this solution into equation (5.2) we obtain the characteristic equation:
λ2 + δ = b e−λT
(5.4)
Note that equation (5.4) is a transcendental equation on λ and therefore has an
infinite number of solutions. If we assume that the roots to the transcendental
equation are complex, then we can write λ = a + iω. Substituting λ = a + iω into
equation (5.4) and solving for the real and imaginary parts of the equation, we
obtain:
a2 − ω 2 + δ = be−aT cos ωT
(5.5)
2aω = be−aT sin ωT
(5.6)
We are interested in knowing for what values of b and T does this system change
stability. The roots of the transcendental characteristic equation hold the stability
84
information. From these roots, we can generate the stability charts (Cf. Figure
5.2).
We know the system is stable when the real part of the root is negative. Thus,
the change of stability occurs when the real part equals zero. Therefore, we set
a = 0 in equations (5.5) and (5.6) to find the values of b and T at the bifurcation
point:
−ω 2 + δ = b cos ωT
(5.7)
0 = b sin ωT
(5.8)
Solving these two equations simultaneously, we retrieve expressions for ω and T
when the real part of the root is zero:
ω 2 = δ − b(−1)n
ωT = nπ,
n = 1, 2, 3, . . .
(5.9)
(5.10)
where cos nπ = (−1)n . Next, solve equation (5.10) for ω and substitute ω into
equation (5.9) to obtain:
δ=
nπ
T
2
+ b(−1)n
(5.11)
which give the stability curves in the δ-b parameter plane for any delay period.
Let’s examine this equation for T = π and T = 2π. The stability charts are
plotted in the top and bottom of Figure 5.4, respectively. Note that the stability
region shrinks as the delay period increases. The first stability region in the top
of Figure 5.4 for T = π is twice as large as the first stability region in the bottom
of Figure 5.4 for T = 2π. Recall that these stability charts were generated based
on the knowledge that an equilibria changes stability when the real part of the
imaginary root goes to zero. When this scenario occurs in ordinary differential
85
Figure 5.4: Stability charts for Hsu-Bhatt equation (5.2) with T=π (top)
and T=2π (bottom). The shaded regions are stable.
equations, a Hopf bifurcation occurs. Since equation (5.2) is linear, only the origin
changes stability when we cross the transition curve.
Up to this point we have only recreated the stability charts of Hsu-Bhatt . We
have obtained the equation that generates the transition curves, equation (5.11).
Let’s examine it’s limit as b → 0 when T = 2π:
δ=
n2
,
4
n = 1, 2, . . .
(5.12)
These values for δ when b = 0 are the same values of δ when ε = 0 in the Mathieu
equation that locate the resonant regions. Thus, the location of the instability
86
regions are the same. This is one possible reason it would be advantageous to
analyze equation (5.1), the linear delayed Mathieu equation, a delay period of
T = 2π . The location of the instability regions in both cases meet at the
same points on the b = 0 and ε = 0 axes, as shown in Figure 5.3. As it turns out,
T = 2π is a special case. In this research effort, we are going to find the stability
charts for a fixed b in the T -δ parameter space for small ε. This study will yield
insight into the effect various delay periods will have on the system.
5.2.2
The Nonlinear Hsu-Bhatt Equation
The nonlinear Hsu-Bhatt equation is a Duffing oscillator combined with delay:
ẍ(t) + δx(t) + εαx(t)3 = bx(t − T )
(5.13)
where α is the amplitude of the cubic nonlinearity. When α = 0, we have the HsuBhatt equation analyzed in section 5.2.1. When b = 0, we have the equation for a
Duffing oscillator, which is a model of a system that includes nonlinear restoring
forces (Cf. Chapter 6). The period-amplitude relationship for the periodic motion
of the Duffing oscillator is given by :
Tpm
3
= 2π 1 − αA2ε + O(ε2 )
8
(5.14)
The nonlinear terms controls the growth of the periodic motion. The period of
the periodic motion grows quadratically with amplitude. In the following section,
we utilize Lindstedt’s perturbation method to investigate what happens to the
period-amplitude relationship when delay is added to the system.
Lindstedt’s Method
Lindstedt’s method is a singular perturbation technique that can be used to derive
the period-amplitude relationship in a nonlinear oscillator, even for delay differ-
87
ential equations . Let’s assume that equation (5.13) with b = 0 has periodic
motion due to the birth of a limit cycle from a Hopf bifurcation. The bifurcation
occurs when the roots to the characteristic equation are purely imaginary, ±iω,
which happens in the b 6= 0 system at a critical delay period, Tcrit . The nonlinear
Hsu-Bhatt equation may exhibit a periodic solution, x(t) = A cos Ωt, for values of
delay period T close to the critical value,
T = Tcrit + εµ + . . .
(5.15)
The main idea behind Lindstedt’s Method is to build the period-amplitude relationship by stretching time:
τ = Ωt,
where
Ω = ω + εk1 + . . .
(5.16)
where k1 is to be found. Since the time dimension of the phase space was stretched,
the delay period needs to be stretched appropriately. The delay term becomes:
x(t − T ) = x(Ω(t − T )) = x(τ − ΩT )
(5.17)
Substituting equations (5.15) and (5.16) into (5.17) we obtain the stretched delay
term:
xd = x(τ − ΩT ) = x(τ − ωTcrit − ε(k1 Tcrit + ωµ))
(5.18)
Next we Taylor expand the delay term, xd , about ε = 0:
xd = x(τ − ωTcrit ) − ε
dx
(τ − ωTcrit) · (k1 Tcrit + ωµ)
dτ
(5.19)
The stretched differential equation comes from substituting equation (5.16) and
(5.19) into equation (6.1):
(ω 2 + 2εωk1 )x(τ )00 + δx(τ ) + εαx(τ )3 =
bx(τ − ωTcrit) − εbx0d (τ − ωTcrit ) · (k1 Tcrit + ωµ)
(5.20)
88
where the prime denotes differentiation with respect to τ . Next we expand x(τ )
in a power series in ε:
x(τ ) = x0(τ ) + εx1(τ ) + . . .
(5.21)
We substitute this expression into (5.20) and collect terms of order epsilon:
ε0 :
ω 2x000 + δx0 = bx0(τ − ωTcrit)
ε1 :
ω 2x001 + δx1 = bx1(τ − ωTcrit)
(5.22)
− 2ωδ1 x000 − αx30 − (δ1 Tcrit + ωµ) · x00d
(5.23)
We take the solution to the ε0−equation as
x0 = A cos τ
(5.24)
We can check that this is the solution by substituting it back into (5.24):
−ω 2 cos τ + δ cos τ = b cos(τ − ωTcrit),
where
ωTcrit = π
(5.25)
resulting in the following expression for ω and Tcrit :
ω 2 = δ + b,
Tcrit = √
π
δ+b
(5.26)
This checks with equation (5.9). Continuing, we substitute x0 = A cos τ into the
ε-equation to obtain the x1-equation:
ω 2 x001 + δx1 − bx1(τ − π) = −2ωδ1(−A cos τ )
−αA
3
1
3
cos τ + cos 3τ − (δ1Tcrit + ωµ)(−A sin(τ − π))
4
4
(5.27)
removal of the resonant terms, cos τ and sin τ , results in two algebraic equations
that we solve to obtain the period-amplitude relationship for the periodic motion:
3
2ωδ1 − αA2 = 0
4
→
A2 =
8ω
δ1
3α
(5.28)
89
δ1Tcrit + ωµ = 0
→
δ1 = −
(δ + b)µ
π
(5.29)
Thus, the period-amplitude relationship is:
3
8(δ + b) 2
µ
A =−
3απ
2
(5.30)
where µ is the detuning from the critical period, Tcrit in equation (5.26), where
the system changes stability. Thus, for a given δ, delay amplitude b, and α > 0,
an unstable limit cycle is born when µ changes from positive to negative, i.e., a
subcritical Hopf bifurcation.
In summary, we presented the stability charts for the Hsu-Bhatt equation for
T = π and T = 2π; we discovered possible motivation as to why Insperger 
generated the stability for the linear delay Mathieu equation for T = 2π, and we
found the period-amplitude relationship for the periodic motion in the nonlinear
Hsu-Bhatt equation. In the following sections, we analyze the behavior of the
nonlinear Hsu-Bhatt equation with 2:1 parametric excitation for a general delay
period, valid for small ε.
5.3
First Order Averaging Method
In preparation for averaging  equation (5.1), we introduce a small parameter ε
by scaling a = εα, b = εβ, and c = εγ. In addition, we detune off of 2:1 resonance
by setting δ =
1
4
+ εδ1:
1
+ εδ1 + εα cos t x(t) + εγx(t)3 = εβx(t − T )
ẍ(t) +
4
(5.31)
When ε = 0, equation (5.31) reduces to ẍ + (1/4)x = 0, with the solution:
x(t) = A cos
t
+φ ,
2
ẋ(t) = −
A
t
sin
+φ
2
2
(5.32)
90
For ε > 0, we look for a solution in the form (5.32) but treat A and φ as time
dependent. Variation of parameters gives the following equations on A(t) and φ(t):
Ȧ(t) = −2ε sin
t
+φ
2
ε
t
+φ
φ̇(t) = −2 cos
A
2
F A cos
A
t
t
+ φ , − sin
+ φ ,t
2
2
2
A
t
t
+ φ , − sin
+ φ ,t
F A cos
2
2
2
(5.33)
(5.34)
where F (x, ẋ, t) = −(δ1 + α cos t)x(t) − γx(t)3 + βx(t − T ) in which x(t) is given
by (5.32). For small ε we use the method of averaging , replacing the righthand sides of (5.33) and (5.34) by their averages over one 2π period of the forcing
function cos t:
1 Z 2π
t
Ȧ ≈ −2ε
+ φ F dt
sin
2π 0
2
Z
ε 1 2π
t
+ φ F dt
φ̇ ≈ −2
cos
A 2π 0
2
(5.35)
(5.36)
in which
t
t
+ φ − γA3 cos3
+φ
F = −(δ1 + α cos t)A cos
2
2
1
+β Ã cos (t − T ) + φ̃
2
(5.37)
where Ã = A(t − T ) and φ̃ = φ(t − T ). Evaluating the integrals in (5.35) and
(5.36) gives the delayed slow flow:
A α sin 2 φ
T
Ȧ = ε
− Ã β sin
− φ̃ + φ
2
2

!
(5.38)

Ã β cos T2 − φ̃ + φ
α cos 2 φ
3γ A2
φ̇ = ε 
−
+
+ δ1 
4
A
2
(5.39)
Equations (5.38) and (5.39) show that Ȧ and φ̇ are O(ε). We now Taylor expand
Ã and φ̃:
1
Ã = A(t − T ) = A(t) − T Ȧ(t) + T 2Ä(t) + . . .
2
(5.40)
1
φ̃ = φ(t − T ) = φ(t) − T φ̇(t) + T 2φ̈(t) − . . .
2
(5.41)
91
Thus we can replace Ã and φ̃ by A(t) and φ(t) in equations (5.38) and (5.39) since
Ȧ and φ̇ and Ä and φ̈ are of O(ε) and O(ε2 ), respectively . This approximation reduces the infinite dimensional problem into a finite dimensional problem by
assuming εT is small.
After substituting the above approximation into (5.38) and (5.39), we obtain
T
α
A =A
sin 2φ − β sin
2
2
0
φ0 =
(5.42)
T
3γ 2 α
A + cos 2φ − β cos + δ1
4
2
2
(5.43)
where primes represent differentiation with respect to slow time η = εt. We may
obtain an alternate form of the slow flow equations (5.42) and (5.43) by transforming from polar coordinates A and φ to rectangular coordinates u and v via
u = A cos φ, v = −A sin φ, giving:
u0 = − β sin
T
2
u + δ1 −
T
α
v = −δ1 − + β cos
2
2
0
T
α
− β cos
2
2
T
u − β sin
2
v+
v−
3γ
v(u2 + v 2)
4
(5.44)
3γ
u(u2 + v 2)
4
(5.45)
Note that from equation (5.32) we have
x(t) = A cos
5.4
t
t
t
t
t
+ φ = A cos φ cos − A sin φ sin = u cos + v sin
(5.46)
2
2
2
2
2
Analysis of the Slow Flow
From equation (5.46) we see that in general an equilibrium point in the slow
flow (5.42) and (5.43) or (5.44) and (5.45) corresponds to a periodic motion in
the original system (5.31). The origin u = v = 0 is an exception since it is an
equilibrium point in both the slow flow equations and in the original system. The
goal is to find nontrivial slow flow equilibria and analyze their stability.
92
5.4.1
Slow Flow Equilibria: Stability and Bifurcation
To find nontrivial slow flow equilibria, we set A0 = φ0 = 0 in equations (5.42) and
(5.43), and utilize the trig identity sin2 2φ + cos2 2φ = 1 to obtain the following
condition on R, where R = A2,
9γ 2 R2 + 24γ δ1 − β cos
T
2
R + 4 4β 2 − α2 + 4δ12 − 8βδ1 cos
T
2
=0
(5.47)
Equation (5.47) is a quadratic on R. The two solutions are:


s
4 
T
1
T
R=
β cos − δ1 ±
α2 − 4β 2 sin2 
3γ
2
2
2
(5.48)
Each value of R corresponds to two nontrivial slow flow equilibria located 180o
apart. This may be seen by noting that equations (5.44) and (5.45) are invariant
under the transformation (u, v) 7→ (−u, −v), which means that if (u, v) is a slow
flow equilibrium, then so is (−u, −v).
R must be nonnegative for nontrivial equilibrium points. This condition yields
two inequalities:
s
1
T
T
δ1 ≤ β cos ±
α2 − 4β 2 sin2
2
2
2
(5.49)
And for real roots, the discriminant in the inequality in (5.49) must be positive.
This condition yields:
T 1 α sin ≤ 2
2 β (5.50)
Hence, for a given α and β, the inequality (5.50) gives a condition on the delay
period T such that there will exist nontrivial slow flow equilibrium points. In the
case that |β/α| ≤ 1/2, there will exist nontrivial fixed points for all T .
When the inequality in (5.50) is satisfied, there are at least two nontrivial slow
flow equilibrium points if,
s
1
T
T
α2 − 4β 2 sin2
δ1 < β cos +
2
2
2
(5.51)
93
In addition, there will be two more nontrivial slow flow equilibrium points if,
s
1
T
T
δ1 < β cos −
α2 − 4β 2 sin2
2
2
2
(5.52)
Thus it is possible to have up to four nontrivial slow flow equilibria. When we
include the origin, this makes a possible total of up to five slow flow equilibria.
Next we investigate which parameter combinations of δ1 and T cause the slow flow
equilibrium points to change stability for a given α and β, and which bifurcations,
if any, accompany the change in stability.
The trace and determinant of the Jacobian matrix evaluated at an equilibrium
point contain the local stability information. Recall that a saddle-node or pitchfork bifurcation generically occurs when the Det(J ) = 0, and a Hopf bifurcation
generically occurs when the T r(J ) = 0 and Det(J ) > 0 . From the slow flow
equations (5.44) and (5.45), the Jacobian matrix is:


J =


J11 J12 

J21 J22
(5.53)

where
J11 = −β sin
3γ
T
+ uv
2
2
J12 = δ1 −
T
3γ
α
9γ
− β cos + u2 + v 2
2
2
4
4
J21 = −δ1 −
T
3γ
α
9γ
+ β cos − v 2 − u2
2
2
4
4
J22 = −β sin
3γ
T
− uv
2
2
where u and v are to be evaluated at the slow flow equilibria.
The trace of the Jacobian matrix is:
T r(J ) = −2β sin
T
2
(5.54)
94
Note that T r(J ) is a function of the delay parameters only, and in particular does
not depend on R. Therefore, T r(J ) = 0 at all of the slow flow equilibrium points
when β = 0, T = 0, or T = 2π. In particular, a change of stability and a possible
Hopf bifurcation (birth of a limit cycle) will occur at T = 2π if Det(J ) > 0.
The determinant of the Jacobian matrix is:
27
T
Det(J ) = γ 2R2 − 3γ β cos − δ1 R
16
2
T
α2 3αγ 2
2
2
− 2βδ1 cos + δ1 + β −
+
(v − u2 )
2
4
4
(5.55)
where the (u2 + v 2) terms were replaced with R. The (v 2 − u2) term is simplified
by multiplying the RHS of equation (5.44) by v and subtracting from it the RHS
of equation (5.45) multiplied by u. This calculation gives:
(v 2 − u2) =
1
T
3γR2 + 4 δ1 − β cos
2α
2
R
(5.56)
Substituting this expression into equation (5.55) we obtain the following expression
for the determinant:
45 2 2 9
T
T
α2
2
2
Det(J ) = γ R − γ β cos − δ1 R − 2βδ1 cos + δ1 + β −
16
2
2
2
4
(5.57)
The value of R at the nontrivial slow flow equilibria are given in equation (5.48).
Substituting that expression for R into the expression for the determinant yields
the value of the determinant at the slow flow equilibria:
T
T
Det(J ) = −4β 2 sin2 + α2 ± 2 β cos − δ1
2
2
s
α2 − 4β 2 sin2
T
2
(5.58)
A change in stability occurs when the determinant vanishes. Thus by setting
equation (5.58) equal to zero, we may solve for a critical value of δ1 , the detuning,
as a function of T for a given α and β:
s
1
T
T
α2 − 4β 2 sin2
δ1 = β cos ±
2
2
2
(5.59)
95
From equation (5.48), equation (5.59) implies R = 0. This means that the stability
change associated with equation (5.59) occurs at the origin. Thus, accompanying
this change in stability is a bifurcation. If the inequalities (5.50), (5.51) and (5.52)
are replaced by equal signs, then (5.50) becomes the condition for a saddle-node
bifurcation and (5.51) and (5.52) become conditions for pitchfork bifurcations. We
illustrate these results by considering two examples.
5.4.2
Example 1
We consider the case in which β =
example becomes:
1
4
and α = γ = 1. Equation (5.59) for this
s
T
1
1
1
T
δ1 = cos ±
1 − sin2
4
2
2
4
2
(5.60)
As stated in the previous section, the origin changes stability along the curves in
the T -δ1 parameter plane given by equation (5.60), corresponding to Det(J ) = 0.
In addition, there is a stability change along the line T = 2π, corresponding to
T r(J ) = 0, when Det(J ) > 0. See Figure 5.5.
Equation (5.60) also corresponds to the occurrence of pitchfork bifurcations in
the slow flow. These curves are shown in the bifurcation set in Figure 5.6. Along
with the curves are points labeled a through i. Each letter corresponds to the
qualitative phase portrait of the respective slow flow system as shown in Figure
5.7. Pitchfork bifurcations occur as we move from top to bottom in each column
of Figure 5.7.
As we move from left to right in each row of Figure 5.7, we cross the line
T = 2π. For those slow flow equilibria for which Det(J ) > 0 (non-saddles), a
stability change is observed, but no limit cycle is seen to be born. The associated
Hopf bifurcation turns out to be degenerate, as will be shown later in the paper.
96
Figure 5.5: Stability of the origin for Example 1, β =
1
4
and α = γ = 1.
Black is stable, white is unstable. The transition curves separating the stability
regions are given by equation (5.60) and by T = 2π.
97
Figure 5.6: Bifurcations in slow flow equilibria for Example 1, β =
1
4
and
α = γ = 1. The letters a, b, c, d, e, f , g, h, i correspond to the qualitative phase
portraits shown in Figure 5.7. Pitchfork bifurcations occur on the Det(J ) = 0
curves, given by equation (5.60). A change of stability occurs on the T r(J ) = 0
curve, T = 2π, in the case that Det(J ) > 0 (non-saddle equilibria). No
limit cycles are born as we cross the latter curve because the associated Hopf
bifurcation is degenerate.
98
Figure 5.7: Qualitative phase portraits for Example 1, β =
1
4
and α = γ = 1.
The letters a, b, c, d, e, f , g, h, i correspond to various parameter combinations
of δ1 and T as shown in Figure 5.6. The bifurcations from a to g, b to h, and
c to i, respectively, are pitchforks. The bifurcations from a to c, d to f , and g
to i, respectively, are degenerate Hopfs.
99
5.4.3
Example 2
We consider the case in which β =
3
5
and α = γ = 1. This example exhibits
qualitatively different behavior from the previous example. Equation (5.59) for
this example becomes:
s
T
1
3
36 2 T
δ1 = cos ±
sin
1−
5
2
2
25
2
(5.61)
Figure 5.8 shows the curves (5.61), corresponding to Det(J ) = 0, which bound
stability regions of the origin. In addition, there is a stability change along the
line T = 2π, corresponding to T r(J ) = 0, when Det(J ) > 0. Note that for
this example, when | sin(T /2)| > 5/6 the discriminant in (5.61) is less than zero,
resulting in a complex conjugate pair. Therefore, it is possible to eliminate the
regions of instability by choosing the delay period T appropriately.
Equation (5.61) also corresponds to the occurrence of pitchfork bifurcations in
the slow flow. These curves are shown in the bifurcation set in Figure 5.9. The
curve T = 2π is also shown in Figure 5.9, representing a degenerate Hopf in which
no limit cycle is born, as in Example 1. In addition, saddle-node bifurcations occur
in Example 2 corresponding to the vanishing of the discriminant in equation (5.61).
These occur when | sin(T /2)| = 5/6, and appear as vertical lines marked Disc = 0
in Figure 5.9.
Along with these bifurcation curves are points labeled a through o. Each letter
corresponds to the qualitative phase portrait of the corresponding slow flow. The
points a through i are qualitatively the same as in Figure 5.7. The points j through
o are shown in Figure 5.10. Saddle-node bifurcations occur as we move from left
to right in each row of Figure 5.10.
100
Figure 5.8: Stability of the origin for Example 2, β =
3
5
and α = γ = 1.
Black is stable, white is unstable. The transition curves separating the stability
regions are given by equation (5.61) and by T = 2π.
101
Figure 5.9: Bifurcation set for the slow flow equilibria for Example 2, β =
3
5
and α = γ = 1. The letters a, b, c, d, e, f , g, h, i correspond to the
qualitative phase portraits shown in Figure 5.7. The letters j, k, l, m, n, o
correspond to the qualitative phase portraits shown in Figure 5.10. Pitchfork
bifurcations occur on the Det(J ) = 0 curves, given by equation (5.61). Saddlenode bifurcations occur on the Disc = 0 curves, given by | sin(T /2)| = 5/6.
A change of stability occurs on the T r(J ) = 0 curve, T = 2π, in the case
that Det(J ) > 0 (non-saddle equilibria). No limit cycles are observed to be
born as we cross the latter curve because the associated Hopf bifurcation is
degenerate.
102
Figure 5.10: Qualitative phase portraits for Example 2, β =
3
5
and α = γ = 1.
The letters j, k, l, m, n, o correspond to various parameter combinations of δ1
and T as shown in Figure 5.9. The bifurcations from j to l and from m to o
are saddle-nodes.
103
5.5
Degenerate Hopf Bifurcation
Limit cycles can be generically created or destroyed in a Hopf bifurcation. The
necessary conditions for a generic Hopf bifurcation in a system of two first order
ODE’s such as the slow flow (5.44) and (5.45) are T r(J ) = 0 and Det(J ) > 0,
where J is the Jacobian matrix evaluated at an equilibrium point. Setting the
trace equal to zero in equation (5.54) for the slow flow gives T = 2π (we ignore
β = T = 0 because the corresponding systems involve no delay). The condition
Det(J ) > 0 corresponds to a non-saddle equilibrium point. Thus if we were to
increase the delay period T through 2π, we would expect to see a limit cycle
created or destroyed (depending upon whether the Hopf was sub - or supercritical)
in the neighborhood of a spiral equilibrium point. See Figure 5.7 where moving
from a to c, from d to f , or from g to i corresponds to increasing the delay period T
through 2π. By inspection no limit cycle is observed, contrary to our expectations.
In the following two subsections, we analyze this behavior by considering the delay
amplitude as is, order ε and we compare it to the case where the delay amplitude
is of order 1, therefore perturbing off of the Hsu-Bhatt equation in equation (5.2)
rather than the simple harmonic oscillator.
5.5.1
Order ε Delay
In order to understand the behavior associated with the degenerate Hopf, we evaluate the slow flow (5.44) and (5.45) at T = 2π:
u
0
v
0
=
=
α
3γ
v(u2 + v 2)
δ1 − + β v +
2
4
α
3γ
u(u2 + v 2)
−δ1 − − β u −
2
4
(5.62)
(5.63)
104
Equations (5.62) and (5.63) possess the following first integral:
2
3γ 2
u + v 2 + 2(δ1 + β) u2 + v 2 + α u2 − v 2 = constant
4
(5.64)
Note that the existence of this first integral for T =2π is in agreement with the phase
portraits b, e and h in Figure 5.7, which were obtained by numerical integration.
In a generic Hopf bifurcation, the limit cycle is born with zero amplitude and
√
grows generically like µ, where µ is the bifurcation parameter. This results in
a family of limit cycles, one for each value of µ. What has happened in the slow
flow (5.44) and (5.45) is that the entire family of periodic motions has occurred
at µ = 0, which corresponds to T = 2π here. The Hopf bifurcation in this case is
degenerate. This phenomenon is well-known , and a condition has been given
which guarantees that no such degeneracy will occur, namely that the equilibrium
point should be a “vague attractor” (or less generally, asymptotically stable) when
µ = 0. Since the slow flow (5.44) and (5.45) has just been shown to be conservative
at T = 2π, this condition does not apply.
Having established that the Hopf bifurcation associated with the slow flow
(5.44) and (5.45) is degenerate, we are led to ask if this nongeneric behavior is
really part of the dynamics of the original equation (5.1), or if it is due to the
nature of the perturbation scheme. In order to investigate this possibility, we
apply an alternative perturbation scheme to equation (5.1), based on taking ε
small in the following equation (Cf. equation (5.31)):
ẍ(t) + δ0x(t) − bx(t − T ) = −ε δ1x(t) + αx(t) cos t + γx(t)3
(5.65)
We treat this equation in the following section, where we use multiple scales to
obtain a slow flow, which we show also exhibits a degenerate Hopf bifurcation.
105
5.5.2
Order 1 Delay
The occurrence of a degenerate bifurcation in a slow flow system derived from a
perturbation method may not be representative of the actual system. The degeneracy may result from the approximation involved in the perturbation method. The
slow flow system (5.44) and (5.45) in the previous section was derived by perturbing off of a simple harmonic oscillator with O(ε) forcing, nonlinearity, detuning,
and delay (Cf. equation (5.31)):
ẍ(t) + δ0x(t) = −ε δ1x(t) + αx(t) cos t + γx(t)3 − βx(t − T )
(5.66)
An alternative approach is to perturb off of the Hsu-Bhatt equation  ẍ + δ0 x =
b(t − T ). We consider a simple harmonic oscillator with a delay term, which is
perturbed by O(ε) forcing, nonlinearity, and detuning :
ẍ(t) + δ0x(t) − bx(t − T ) = −ε δ1x(t) + αx(t) cos t + γx(t)3
(5.67)
where the delay amplitude is b, the delay period is T and δ0 is the frequency
squared of the simple harmonic oscillator. If the delay amplitude b and ε were
zero, then this system would be stable when δ0 > 0 and unstable when δ0 < 0. By
changing the delay parameters b and T , the stable system can become unstable and
the unstable system can become stable. Thus a change of stability would occur.
From section 5.2.1, we know the solution is of the form:
x(t) = eλt
(5.68)
where the characteristic equation is:
λ2 + δ0 = b e−λT
(5.69)
Note that equation (5.69) is a transcendental equation on λ and therefore has an
infinite number of (generally complex) roots. For a Hopf bifurcation, a pair of
106
roots must be pure imaginary. Substituting λ = iω into equation (5.69) yields:
2
ωcrit
= δ0 + b
ωcrit Tcrit = π
(5.70)
which gives the solution to the Hsu-Bhatt equation (5.67) for ε = 0 right at a Hopf
bifurcation point. This is the starting point for the perturbation method.
We use the multiple time scales method and begin by perturbing off of the
critical value of the delay period:
T = Tcrit + εµ + . . .
(5.71)
We then define two time scales ξ = t and η = εt, and expand x in a power series of
ε: x(ξ, η) = x0 (ξ, η) + εx1 (ξ, η) + . . . We substitute these expansions into equation
(5.67) and collect like terms. The solution to the ε = 0 equation is:
x0 (ξ, η) = A1(η) cos ωcrit ξ + A2(η) sin ωcrit ξ
(5.72)
where A1 (η) and A2(η) are functions of slow time η. In order for the cos t term in
equation (5.67) to be 2:1 resonant, ωcrit = 12 . This implies from equation (5.70):
δ0 + b =
1
4
Tcrit = 2π + εµ
(5.73)
The O(ε) equation is:
x1ξξ + δ0x1 − bx1(ξ − Tcrit) =
− 2x0ξη + (δ1 + α cos ξ)x0 + γx30 + µx0ξ (ξ − Tcrit)
(5.74)
where the subscripts denote partial derivatives and x0 is given by equation (5.72).
In deriving equation (5.74) the product εTcrit has been assumed to be small .
After substituting equation (5.72) into (5.74), the removal of resonant terms yields
107
the slow flow:
dA1
bµ
α
3γ
= A1 + δ1 −
A2 + A2(A21 + A22 )
dη
2
2
4
dA2
bµ
α
3γ
= A2 − δ1 +
A1 − A1(A21 + A22 )
dη
2
2
4
(5.75)
(5.76)
As stated in section 5.5.1, the necessary conditions for a generic Hopf bifurcation
in a system of two first order ODE’s, such as those above in (5.75) and (5.76),
are T r(J ) = 0 and Det(J ) > 0, where J is the Jacobian matrix evaluated at
the equilibrium point. The trace and determinant for the slow flow system above
about the origin are:
T r(J ) = bµ
Det(J ) =
b2 µ2
α
+ δ12 −
4
4
(5.77)
Setting the T r(J ) = 0 gives b = 0 (which corresponds to a system with no delay)
or µ = 0, which is right at the bifurcation point of Tcrit = 2π. Using µ = 0, the
condition for the Det(J ) > 0 becomes:
δ12 >
α
4
⇒
|δ1| >
α
2
(5.78)
Thus, for a given α > 0, and choosing |δ1| > α/2, we would expect a limit cycle to
be born as we change µ from negative to positive or positive to negative. However,
a limit cycle is not born in this slow flow system, contrary to what we expect. In
fact, the slow flow in (5.75) and (5.76) at the bifurcation point of µ = 0 has the
first integral:
6γA22 + 8δ1 + 4α
A2 3γ 4
A21
+ (8δ1 − 4α) 2 +
A1 + A42 = C
4
4
4
(5.79)
The existence of this first integral signifies that the Hopf bifurcation is degenerate,
as explained at the end of section 5.5.1.
108
Thus both perturbation schemes, one in which the delay amplitude is O(1)
and the other in which the delay amplitude is O(ε), give slow flow systems which
exhibit degenerate Hopf bifurcations.
5.5.3
Numerical Investigation
Further evidence of the existence of the degenerate Hopf in the original equation
(5.1) was obtained by numerically integrating (5.1) in the neighborhood of T = 2π
for a variety of other parameters. Since a limit cycle in the slow flow corresponds
to a quasiperiodic motion in the original equation, we searched for quasiperiodic
motions in (5.1). No quasiperiodic motions were observed.
5.6
Comparison with Numerical Integration
In this section we compare the foregoing analytical results based on the slow flow
(5.42) and (5.43) or (5.44) and (5.45), with direct numerical integration of the
delayed Mathieu equation (5.31). The numerical integration was completed in
MATLAB, which has recently extended the differential equation package to numerically integrate systems of delay differential equations . For this task, we
used the integrating function dde23. There is an on-line tutorial available at MATLAB Central .
The results are for the stability of the origin, x = ẋ = 0, and are shown in
Figures 5.11 and 5.12. The same parameters were used as in Examples 1 and
2, except we took γ = 0, since the stability of the origin does not involve the
nonlinear term in equation (5.31). In particular we used β = 0.25 in Figure 5.11
and β = 0.60 in Figure 5.12. A MATLAB script was written to generate these
figures numerically, utilizing the dde23 function. The T -δ1 parameter plane was
109
divided into a grid of 10,000 points. A point in this space is deemed stable if,
after 2000 forcing periods starting with the initial condition x = 1, the norm of
the amplitude is less than 1010 , otherwise it is unstable. For each figure, ε = 0.05.
In Figure 5.11, we note the appearance of a stable region on the right side of
the numerical (lower) stability chart which is absent from the analytical (upper)
stability chart. This may be explained by recalling that the analytical results are
based on a perturbation method which assumed that εT was small.
110
Figure 5.11: Stability of the origin for β = 0.25 and α = 1. Comparison
between analytical result based on slow flow (upper) with numerical integration
of equation (5.31) for ε = 0.05 (lower). Black is stable, white is unstable.
111
Figure 5.12: Stability of the origin for β = 0.60 and α = 1. Comparison
between analytical result based on slow flow (upper) with numerical integration
of equation (5.31) for ε = 0.05 (lower). Black is stable, white is unstable.
112
5.7
Discussion
The main difference between Examples 1 and 2 presented earlier in this chapter
is that Example 2 involves a saddle-node bifurcation and Example 1 does not.
The saddle-node bifurcation occurs because the condition for nontrivial slow flow
equilibria to exist, equation (5.50), is not satisfied for all values of T . Recall that
the instability region in the T -δ1 parameter plane was bounded by curves with
equations (5.51) and (5.52). Thus, for each delay period T , and for given values of
α and β, there are two critical values of δ1 if β < α/2. If β > α/2, however, there
are values of T for which there are no real critical values of δ1 , and therefore there
is no instability.
Recalling that δ = 1/4 + δ1ε, we may write the transition curves separating
stable from unstable regions in the form:

δ=
s

1
T
1
T
+ ε β cos ±
α2 − 4β 2 sin2 
4
2
2
2
(5.80)
Figure 5.13 shows the instability region (5.80) in the three dimensional δ-T -ε parameter space for the parameters of Example 2. Figure 5.13 can be compared with
Figure 5.12 by intersecting the three dimensional instability tongue with the plane
ε = 0.05. Then the white region lying inside the tongue in Figure 5.13 is the same
as the white instability region on the left side of Figure 5.12.
In the limiting case of β = 0 (no delay term), equation (5.1) becomes a nonlinear
Mathieu equation, the properties of which are well-known  and . See Figure
5.14 which displays a bifurcation diagram for equation (5.1) with α = γ = 1, β = 0.
The effect of adding a small delay term may be understood by perusing Figures 5.6
and 5.7. These show, first of all, that the presence of delay introduces dissipation
113
into the slow flow. Moreover we see that if the delay amplitude β is small enough
(0 < β < α/2), and if the delay period T is small enough (T < 2π), then the
dynamics of the delayed equation is similar to that of a damped nonlinear Mathieu
equation without delay. If, however, the delay amplitude is large enough (β > α/2),
then we have seen that it is possible to eliminate the tongue of instability by
choosing the delay period appropriately. Figure 5.13 shows the tongue closing as
T → T = 2 sin−1
α
.
2β
See also Figures 5.9 and 5.10.
We have seen that the stability change at the origin is given by equation (5.59).
This equation can be rewritten so as to give the following condition for instability
at the origin:
δ12 + β 2 − 2βδ1 cos
α2
T
−
<0
2
4
(5.81)
This shows that the origin is unstable for all delay periods T if the delay amplitude
β and the detuning δ1 are taken to be sufficiently small. For example, in Figure
5.5, where β = 14 , the origin is always unstable for |δ1| < 14 .
5.8
Conclusions
In this chapter, we investigated the dynamics of equation (5.1) which involves
the interaction of parametric excitation with delay. Our analytical results are
based on slow flow equations (5.44) and (5.45) which were obtained by use of
first order averaging. We studied the stability of the origin and the bifurcations
which accompanied stability changes: pitchforks, saddle-nodes, and degenerate
Hopf bifurcations.
We showed that adding delay to an undamped parametrically excited system
introduces effective damping. Our most important conclusion is that for sufficiently
large delay amplitudes β, and for appropriately chosen delay periods T , the 2:1
114
instability region associated with parametric excitation can be eliminated. This
result has potential utility in applications where instabilities are undesirable.
Figure 5.13: The 2:1 instability tongue (5.80), for β = 35 , α = γ = 1. U is
unstable, S is stable. The white instability region lying in the plane ε = 0.05
corresponds to the white instability region on the LHS of Figure 5.12.
115
Figure 5.14: Bifurcation diagram for the nonlinear Mathieu equation without
delay term, equation (5.1) with α = γ = 1, β = 0.
Chapter 6
Conclusions and Future Work
“Science is built up of facts, as a house is with stones. But a collection
of facts is no more a science than a heap of stones is a house.”
– Henri Poincaré
Parametric resonance is a phenomenon that can both stabilize and destablize a
system. It causes large instabilities when the excitation is applied at twice the
system’s natural frequency. In this thesis, we examined how a system responded to
2:1 parametric excitation in the presence of a second parametric driver (Chapter
3), in the presence of a Hopf bifurcation (Chapter 4), and in the presence of
nonlinearities and delay (Chapter 5). In this final chapter, we offer some ideas for
future work which would be an extension of the research conducted in this thesis.
In Chapter 6, we learned that delay adds dissipation to a system for certain
delay periods and amplitudes. In Chapter 4, we showed that a system with linear
damping and cubic nonlinearities experiences a Hopf bifurcation when the linear
damping is tuned from positive to negative. Hence, a possible research effort would
116
117
be to investigate the response of a system with 2:1 parametric excitation, cubic
nonlinearities, linear damping and delay, focusing on the relationship between the
delay and damping parameters.
In Chapter 2, we showed that the Mathieu equation models a system with 2:1
parametric resonance. A second possible research effort would involve analyzing
the stability in a Mathieu-like system where the damping coefficient is a periodic
function , , . Then, consider the effect delay has on the transition curves
in the δ-ε plane.
In Chapter 5, we showed that a Hopf bifurcation occurs in the unforced nonlinear system when A = 0, where A is the linear damping coefficient. (Note the
a saddle-connection bifurcation occurs in the unforced system when A = −1; this
was not shown.) We then analyzed the stability of the steady state solution in
the presence of 2:1 parametric excitation by detuning off of the resonance with κ.
We found that the steady state behavior changed from periodic to quasiperiodic
motion for A < 0. A third possible research effort would be to examine what
happens to that quasiperiodic steady state as the bifurcation parameters A and
κ proceed towards the saddle-connection bifurcation at A = −1. In the following
sections, we provide the foundation for starting that investigation.
6.1
The Duffing Oscillator
The nonlinear constant coefficient second order differential equation
ẍ + x + αx3 = 0
(6.1)
is well known as the Duffing oscillator, where αx3 is a positive restoring force when
α > 0 or a negative restoring force when α < 0 . Let’s consider the long term
118
behavior of this system. The flow in the phase plane for a given initial condition
satisfies the following differential equation,
−αx3 − x
dy
=
dx
y
(6.2)
y 2 x2 αx4
+
+
= constant
2
2
4
(6.3)
where the solution is:
The phase plane is filled with these level curves given by equation (6.3). Along
each trajectory, the quantity above is conserved.
To further analyze the system, we find the equilibrium points to equation (6.2),
which are dependent on the sign of the restoring force, α. When α > 0, the origin
is the only equilibrium point and the solution is a continuum of level curves that
are periodic, shown in Figure 6.1a. However, when α < 0, there are two nontrivial
√
equilibria: x∗ = ±1/ −α. Therefore, all initial conditions starting near the origin
are periodic and those away from the origin tend to infinity. The curve that
separates the periodic from the unbounded solutions is called the separatrix.
After numerically integrating Duffing’s equation, we find that the period of
the solutions inside the separatrix are amplitude dependent. This behavior is not
evident from the phase plane analysis. In the next section, we use elliptic functions
to find the period-amplitude relationship for the Duffing Oscillator.
119
Figure 6.1: The phase plane analysis of the Duffing Oscillator with (a) positive nonlinear restoring forces and (b) negative nonlinear restoring forces.
6.1.1
Elliptic Function Solution
Let’s consider the Duffing oscillator with a negative nonlinear restoring force. This
form of the equation is also known as the Escape equation :
ẍ + x − x3 = 0
(6.4)
Let’s assume the solution to equation (6.4) is a combination of elliptic functions
,
x(t) = a1
cn(a2t, k)
dn(a2 t, k)
(6.5)
where a1 is the amplitude of oscillation, a2t is the amplitude, and k is the modulus.
We solve for a2 and k by substituting x(t) into the equation of motion (6.4) and
collecting like terms. In order to make that substitution, we need an expression
for ẍ:
ẋ(t) = a1 a2(k 2 − 1)
sn(a2t, k)
dn2 (a2t, k)
ẍ(t) = a1 a22(k 2 − 1)(1 + k 2 − k 2 cn(a2t, k))
(6.6)
cn(a2t, k)
dn3 (a2t, k)
120
Substituting ẍ into equation (6.4), multiplying by dn3 , and utilizing elliptic function identities, we obtain an algebraic expression on a1, a2 and k:
a1 a22k 2 + a1k 2 − a1a22 k 4 − a31 cn3 + a1a22 k 4 − a1 k 2 − a1 a22 + a1 cn = 0
(6.7)
When we set the coefficients of cn and cn3 to zero to satisfy equation (6.7), we
retrieve two algebraic equations on a2 and k. Solving those, we obtain
k2 =
a21
2 − a21
a22
2 − a21
2
=
!
!
(6.8)
(6.9)
Thus, the periodic solution to the Duffing equation (6.4) for α < 0 is equation
(6.5), where a2 and k are given above. We also obtain from the analysis the period
of oscillation, T :
T =
4K(k 2 )
a2
(6.10)
where K is the complete elliptic integral of the first kind.
6.2
The Forced Duffing Oscillator
Next, let’s consider the Duffing oscillator with a negative nonlinear restoring force
(α < 0) and a small perturbing force:
z̈ + z − z 3 = −ε(Aż + ż 3)
(6.11)
where ε << 1 and A is linear damping. We use regular perturbations to investigate
the dynamics of this equation by expanding z as a power series in ε, z = z0 + εz1 +
O(ε2 ) in equation (6.11),
z̈0 + εz̈1 + z0 + εz1 − (z03 + 3εz02 z1) = −ε(Aż0 + ż03 )
(6.12)
121
Collecting orders of ε yields:
ε0 :
z̈0 + z0 − z03 = 0
(6.13)
ε1 :
z̈1 + z1 − 3z02 z1 = −(Az˙0 + z˙03 )
(6.14)
The zeroth order equation (6.13) is the Duffing equation we solved in the previous
section. Thus, the zeroth order solution is:
cn(a2t, k)
z0(t) = a1
,
dn(a2t, k)

1

,
k = ±a1  q
2 − a21
a2 =
v
u
u
t
2 − a21
2
!
(6.15)
where a1 is the amplitude of the periodic motion. Now we are able to solve the z1
equation (6.14) and find the relationship between the damping coefficient A and
the amplitude of the periodic motion, a1.
The z1 differential equation is a linear differential equation with coefficients
that are elliptic functions. Since it’s an ODE, it has a complementary and particular solution. One part of the complementary solution is found by solving the
homogeneous equation:
z̈1 + z1 − 3z02 z1 = 0
(6.16)
which can also be solved by differentiating the zeroth order equation with respect
to time:
i
d h
z̈0 + z0 − z03 = 0
dt
⇒
d3 z0 dz0
− 3z02 ż0 = 0
+
3
dt
dt
(6.17)
By comparing the above expression with equation (6.16), we see that z1 = C1 ż0
is the periodic solution. Since equation (6.14) is a second order differential equation, we expect the complementary solution to consist of two linearly independent
solutions:
zc (t) = C1 z˙0 + C2 f (t)
(6.18)
122
where f (t) is not a periodic function. Thus, for the Duffing oscillator, we only
consider the periodic term of the complementary solution.
Next, let’s consider the periodic solution to the nonhomogeneous equation.
Since the terms on the RHS of equation (6.14) are not trigonometric functions
(where the removal of resonant terms is straightforward) but elliptic functions,
we use the Fredholm Alternative Theorem to find the relationship between the
damping coefficient A and the amplitude of the periodic motion a1.
The Fredholm Alternative Theorem
The Fredholm Alternative Theorem states that in order for a solution to exist, the
RHS of the nonhomogeneous equation must be orthogonal to the null space of the
adjoint operator of the homogeneous equation.
Recall that given an operator L, the adjoint operator L∗ is defined by:
(u, Lv) = (v, L∗u)
(6.19)
where the notation (, ) refers to the inner product. The linear differential operator for the z1 equation is,
d2 L() = 2 + − 3z02 ()
dt
(6.20)
The adjoint operator is found by solving equation (6.19) using the operator given
above, and utilizing integration by parts:
(u, Lv) =
Z
00
u [v + v −
3z02v]
dt =
Z
v [u00 + u − 3z02 u] dt = (v, L∗u)
(6.21)
Thus, L(v) = L∗ (u), i.e., the linear differential operator is self-adjoint. Therefore,
the null space of the adjoint operator is the solution to the z1 homogeneous equation. This means, the RHS of the z1 nonhomogeneous equation must be orthogonal
123
to the periodic solution of the z1 homogeneous equation. That is to say, A must
satisfy:
Z
T
0
z˙0 (Az˙0 + z˙03 )dt = 0
(6.22)
where T is one period of z0 . Let’s simplify this equation by switching the integration variables from t to u = a2t. Hence, the limits of integration change to 0 and
4K. Therefore, equation (6.22) becomes,
I
A
sn2(u, k)
du +
dn4(u, k)
I
a21a22 (k 2 − 1)2
sn4(u, k)
du = 0
dn8 (u, k)
(6.23)
Solving the above equation for A, we obtain,
A=−
H
sn4 (u,k) du
8
dn (u,k)
H sn2 (u,k)
du
dn4 (u,k)
a21a22(k 2 − 1)2
(6.24)
The solution to equation (6.24) yields the relationship between the amplitude of
the periodic motion a1 and the damping coefficient A. In order get an expression
for A as a function of a1 , we need to compute integrals of elliptic functions.
Byrd and Friedman’s Handbook of Elliptic Integrals  contains formulas to
compute derivatives and integrals of elliptic functions. There are two integrals of
elliptic functions that we need to compute. Since these two integrands differ only
by an exponent, we can use the same formula to calculate them both: equation
351.51 on page 209 in Byrd and Friedman,
Z
sn2m u cn2n u nd2p u du
(6.25)
where nd = 1/dn. For the integral in the numerator of equation (6.24), m =
2, n = 0, p = 2 and for the integral in the denominator m = 1, n = 0, p = 2.
The solution to these integrals are found using the following recursive formula:
Z
sn2m u cn2n u nd2pu du =
124
1
k 2(m+n)
n
m X
X
j=0 l=0
(−1)j+l+n k 02(n−l) m! n!
I2(p−j−l) (6.26)
(m − j)! j! (n − l)! l!
√
where k 0 = 1 − k 2 is the complementary modulus. The formula for I2(p−j−l)
refers to equation 315 on page 195. After these substitutions, the integrals in the
numerator and denominator of equation (6.24) simplify to,
1
(I8 − 2 I6 + I4)
k4
1
= 2 (I4 − I2)
k
Inum =
(6.27)
Idem
(6.28)
where the Ij are given by:
I2 =
1
E
k 02
I4 =
1 2
02
4E
−
2k
E
−
4Kk
3k 04
I6 =
1 2
4(2
−
k
)I
−
3I
4
2
5k 02
I8 =
1 2
6(2
−
k
)I
−
5I
6
4
7k 02
(6.29)
and where K and E are the complete elliptic integrals of the first and second
kind, respectively. Using MACSYMA to solve the integrals in equation (6.24),
we obtain a relationship between amplitude of the periodic motion, a1 and the
damping coefficient, A and a relationship between the amplitude of the periodic
motion and it’s period, T in equation (6.10). These relationships are plotted in
Figure 6.2.
First note that as A → Acrit ≈ −0.34 , the amplitude a1 → 1, which is
the location of the saddle point when α = −1 from equation (6.1), see Figure
6.2a. Second, note that as the amplitude of the periodic motion goes to one, the
period approaches infinity, see Figure 6.2b. Thus, the elliptic functions formulation
is a good approximation to the lightly damped Duffing oscillator with negative
restoring force since it predicts that as a1 → 1, k 2 → ∞. (Cf. Figure 6.1b.)
125
Figure 6.2: (a) The amplitude of the periodic motion as a function of damping. (b) The period as function of the amplitude.
126
6.3
The Duffing Oscillator with 2:1 Parametric
Excitation
To investigate how the quasiperiodic steady state behavior of Chapter 5 is affected by 2:1 parametric excitation, let’s consider the Duffing oscillator with 2:1
parametric excitation and small perturbing force:
ẍ + x − x3 + ε(k1 + B cos 2t)x = −ε(Aẋ + ẋ3)
(6.30)
where A is the linear damping, B is the amplitude of the parametric resonance and
k1 is the detuning off of the 2:1 resonance. Approaching the problem as outlined
in the previous section, we obtained an expression for A that included integrals of
elliptic functions, trigonometric functions and the parameters B and a1 .
A recent revelation by Rand leads to a possible resolution. He showed in 
that instead of taking the solution to the Duffing oscillator (6.4) as equation (6.5),
it can be rewritten more simply as
z(t) = a1 sn(a2t + b, k)
(6.31)
This knowledge would greatly simplify the previous analysis using elliptic functions, and would lead to alternative methods for solving the integrals of elliptic
functions. Equation (6.31) should be used in the future investigations of this work.
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