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- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
The book presents a unified view on nonlinear systems in physics by providing a common framework to obtain approximate solutions to the respective nonlinear partial differential equations based on the asymptotic perturbation method.
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Veröffentlichungsjahr: 2023
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Table of Contents
Cover
Title Page
Copyright
About the Author
Foreword
Introduction
1 The Asymptotic Perturbation Method for Nonlinear Oscillators
1.1 Introduction
1.2 Nonlinear Dynamical Systems
1.3 The Approximate Solution
1.4 Comparison with the Results of the Numerical Integration
1.5 External Excitation in Resonance with the Oscillator
1.6 Conclusion
2 The Asymptotic Perturbation Method for Remarkable Nonlinear Systems
2.1 Introduction
2.2 Periodic Solutions and Their Stability
2.3 Global Analysis of the Model System
2.4 Infinite-period Symmetric Homoclinic Bifurcation
2.5 A Few Considerations
2.6 A Peculiar Quasiperiodic Attractor
2.7 Building an Approximate Solution
2.8 Results from Numerical Simulation
2.9 Conclusion
3 The Asymptotic Perturbation Method for Vibration Control with Time-delay State Feedback
3.1 Introduction
3.2 Time-delay State Feedback
3.3 The Perturbation Method
3.4 Stability Analysis and Parametric Resonance Control
3.5 Suppression of the Two-period Quasiperiodic Motion
3.6 Vibration Control for Other Nonlinear Systems
4 The Asymptotic Perturbation Method for Vibration Control by Nonlocal Dynamics
4.1 Introduction
4.2 Vibration Control for the van der Pol Equation
4.3 Stability Analysis and Parametric Resonance Control
4.4 Suppression of the Two-period Quasiperiodic Motion
4.5 Conclusion
5 The Asymptotic Perturbation Method for Nonlinear Continuous Systems
5.1 Introduction
5.2 The Approximate Solution for the Primary Resonance of the
n
th Mode
5.3 The Approximate Solution for the Subharmonic Resonance of Order One-half of the
n
th Mode
5.4 Conclusion
6 The Asymptotic Perturbation Method for Dispersive Nonlinear Partial Differential Equations
6.1 Introduction
6.2 Model Nonlinear PDES Obtained from the Kadomtsev–Petviashvili Equation
6.3 The Lax Pair for the Model Nonlinear PDE
6.4 A Few Remarks
6.5 A Generalized Hirota Equation in 2 + 1 Dimensions
6.6 Model Nonlinear PDEs Obtained from the KP Equation
6.7 The Lax Pair for the Hirota–Maccari Equation
6.8 Conclusion
7 The Asymptotic Perturbation Method for Physics Problems
7.1 Introduction
7.2 Derivation of the Model System
7.3 Integrability of the Model System of Equations
7.4 Exact Solutions for the C-integrable Model Equation
7.5 Conclusion
8 The Asymptotic Perturbation Model for Elementary Particle Physics
8.1 Introduction
8.2 Derivation of the Model System
8.3 Integrability of the Model System of Equations
8.4 Exact Solutions for the
C
-integrable Model Equation
8.5 A Few Considerations
8.6 Hidden Symmetry Models
8.7 Derivation of the Model System
8.8 Coherent Solutions
8.9 Chaotic and Fractal Solutions
8.10 Conclusion
9 The Asymptotic Perturbation Method for Rogue Waves
9.1 Introduction
9.2 The Mathematical Framework
9.3 The Maccari System
9.4 Rogue Wave Physical Explanation According to the Maccari System and Blowing Solutions
9.5 Conclusion
10 The Asymptotic Perturbation Method for Fractal and Chaotic Solutions
10.1 Introduction
10.2 A New Integrable System from the Dispersive Long-wave Equation
10.3 Nonlinear Coherent Solutions
10.4 Chaotic and Fractal Solutions
10.5 Conclusion
11 The Asymptotic Perturbation Method for Nonlinear Relativistic and Quantum Physics
11.1 Introduction
11.2 The NLS Equation for
a
1
> 0
11.3 The NLS Equation for
a
1
< 0
11.4 A Possible Extension
11.5 The Nonrelativistic Case
11.6 The Relativistic Case
11.7 Conclusion
12 Cosmology
12.1 Introduction
12.2 A New Field Equation
12.3 Exact Solution in the Robertson–Walker Metrics
12.4 Entropy Production
12.5 Conclusion
13 Confinement and Asymptotic Freedom in a Purely Geometric Framework
13.1 Introduction
13.2 The Uncertainty Principle
13.3 Confinement and Asymptotic Freedom for the Strong Interaction
13.4 The Motion of a Light Ray Into a Hadron
13.5 Conclusion
14 The Asymptotic Perturbation Method for a Reverse Infinite-Period Bifurcation in the Nonlinear Schrodinger Equation
14.1 Introduction
14.2 Building an Approximate Solution
14.3 A Reverse Infinite-Period Bifurcation
14.4 Conclusion
Conclusion
References
Index
End User License Agreement
List of Illustrations
Chapter 1
Figure 1.1 Dissipation (
α
1
)–response (
ρ
E
) space. Rectangles are st...
Figure 1.2 Phase space diagram (
X
(
t
),
Y
(
t
)) with
with f2 = 2, f3 = 1, g0 =...
Figure 1.3 Phase space diagram (
X
(
t
),
Y
(
t
)) with
and f2 = 2, f3 = 1, g0 = ...
Figure 1.4 External force–response space. Circles are sources.
Figure 1.5 External force–response space. Rectangles are sinks.
Figure 1.6 External force–response space. Rectangles are sinks, crosses repr...
Figure 1.7 Associated map of the nonautonomous Eq. (1.27) with f2 = −1, f3 ...
Chapter 2
Figure 2.1 Response–parametric excitation curve, i.e. amplitude of the respo...
Figure 2.2 Frequency–response curve, i.e. the response (
ρ
) as function ...
Figure 2.3 Phase-space diagram (
X
(
t
),
Y
(
t
)) with
with f2 = 1.0, f3 = −0.1...
Figure 2.4 Phase-space diagram (
X
(
t
),
Y
(
t
)) with
and f2 = 1.0, f3 = −0.11...
Figure 2.5 Representation in the
X–Y
space of an orbit with f2 = 1.0, ...
Figure 2.6 Associated map of the Liénard system (2.4) with f2 = 1.0, f3 = 0....
Figure 2.7 A two-period quasiperiodic motion for the Liénard system with α1 ...
Figure 2.8 Phase-space diagram (
X
=
ρ
cos
θ
a...
Figure 2.9 Phase-space diagram (
X
=
ρ
cos
J
and Y ...
Figure 2.10 A periodic motion for the Liénard oscillator, corresponding to t...
Figure 2.11a Phase-space diagram (X = ρ cosθ...
Figure 2.11b A two-period quasiperiodic motion for the Liénard oscillator, c...
Figure 2.12a Phase-space diagram (
X
=
ρ
cos
θ
Figure 2.12b A two-period quasiperiodic motion for the Liénard oscillator, c...
Figure 2.12c Fourier spectrum of the motion in Figure 2.12b. No windows were...
Figure 2.12d Fourier spectrum of the motion in Figure 2.12b; 310 lines of re...
Figure 2.13a Phase-space diagram (
X
=
ρ
cos
θ
Figure 2.13b A two-period quasiperiodic motion for the Liénard oscillator co...
Figure 2.14 The quasiperiodic forcing with Ω = 1.618033989…,
K
= 5, and
ω
...
Figure 2.15 The attractor Poincarè section with initial conditions
X
= 2.0 a...
Figure 2.16 The attractor Poincarè section with initial conditions
X
= 2.0 a...
Figure 2.17 The attractor Poincarè section with initial conditions
X
= 2.0 a...
Figure 2.18 The attractor Poincarè section with initial conditions
X
= 2.0 a...
Figure 2.19 The attractor Poincarè section with initial conditions
X
= 1.0 a...
Figure 2.20 The attractor Poincarè section with initial conditions
X
= 1.1 a...
Figure 2.21 The attractor Poincarè section with initial conditions
X
= 1.0 a...
Figure 2.22 The attractor Poincarè section with initial conditions
X
= 1.2 a...
Figure 2.23 The attractor Poincarè section with initial conditions
X
= 0.7 a...
Figure 2.24 Pictorial three-dimensional solution representation of the Figur...
Chapter 3
Figure 3.1a Parametric excitation (
F
)–response (
ρ
) curves for the uncon...
Figure 3.1b Parametric excitation (
F
)–response (
ρ
) curves for the contr...
Figure 3.1c Parametric excitation (
F
)–response (
ρ
) curves for the syste...
Figure 3.1d Parametric excitation (
F
)–response (
ρ
) curves for the syste...
Figure 3.2 Frequency (
s
)–response (
ρ
) curves for the uncontrolled syste...
Figure 3.3 Parametric excitation (
F
)–response (
ρ
) curves for the contro...
Chapter 4
Figure 4.1 Parametric excitation (
f
)–response (
ρ
) curves for the uncont...
Figure 4.2 Parametric excitation (
f
)–zero-mode amplitude (
χ
) curves for...
Figure 4.3 Parametric excitation (
f
)–response (
ρ
) curves for the uncont...
Figure 4.4 Parametric excitation (
f
)–zero-amplitude response (
χ
) curves...
Figure 4.5 Frequency (
s
)–response (
ρ
) curves for the uncontrolled syste...
Chapter 7
Figure 7.1 A soliton solution at time
T
=0 (with
A
1
= 0.3, α1,1...
Figure 7.2 Evolution plots of two dromions with different shapes and amplitu...
Figure 7.3 A lump solution (with
A
1
= 0.5,
B
1
= 0.3,
C
1
= 0.3, K1, 1 = 0.6...
Figure 7.4 A ring soliton (a) with
A
1
= 1,
B
1
= 0.6, R0,1 = 140...
Figure 7.5 A special instanton solution (with
A
1
= 1, α1,1 = 0.1...
Figure 7.6 A moving breather-like solution (with
A
1
= 1,
B
1
= 0.6, α1,1 = 0....
Chapter 8
Figure 8.1 Evolution plots of two dromions with different shapes and amplitu...
Figure 8.2 Evolution plots of two dromions with identical shapes and amplitu...
Figure 8.3 A ring soliton (solution (8.90a) with
M
= 100 GeV/c
Figure 8.4 An amplitude chaotic–chaotic pattern at
t
= 0 (see solution (8.95...
Figure 8.5 A stochastic fractal solution with the Weierstrass function at
t
...
Chapter 9
Figure 9.1 The group velocity
V
2
as a function of
K
1
when the condition (9.3...
Chapter 10
Figure 10.1 Evolution plots of two dromions given by the solution (10.52) (R...
Figure 10.2 An amplitude chaotic–dromion pattern at
t
= 0 (see solution (10....
Figure 10.3 A stochastic fractal solution with the Weierstrass function at
t
Chapter 11
Figure 11.1 Contour plot for the relative shift (11.66) with
X
= quantum num...
Chapter 12
Figure 12.1 Luminosity distances as a function of the redshift. The crosses ...
Chapter 14
Figure 14.1 Frequency-response curve for the focusing case (
f
= 0.08). The u...
Figure 14.2 Frequency response curve for the defocusing case (
f
= 0.08). The...
Figure 14.3 Frequency near the bifurcation point (
f
C
= 0.02). The behavior f...
Figure 14.4 Level curves for the function
E(ρ, ϑ)
in the focusing ...
Guide
Cover
Table of Contents
Title Page
Copyright
About the Author
Foreword
Introduction
Begin Reading
Conclusion
References
Index
End User License Agreement
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Asymptotic Perturbation Methods
For Nonlinear Differential Equations in Physics
Attilio Maccari
Author
Prof. Attilio MaccariVia Alfredo Casella 300013 RomeItaly
Cover Image: © Callista Images/Getty Images
All books published by WILEY-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.
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A catalogue record for this book is available from the British Library.
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The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at <http://dnb.d-nb.de>.
© 2023 WILEY-VCH GmbH, Boschstraße 12, 69469 Weinheim, Germany
All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law.
Print ISBN: 978-3-527-41421-5ePDF ISBN: 978-3-527-84172-1ePub ISBN: 978-3-527-84173-8oBook ISBN: 978-3-527-84174-5
About the Author
Attilio Maccari is a free-lance researcher in nonlinear physics. He received his PhD from La Sapienza Rome University in 1990. He has published about one hundred papers, mainly on coupled oscillators and nonlinear partial differential equations in physics and engineering.
Among his most important accomplishments, we recall the Maccari system for rogue waves as well as the Hirota–Maccari equation for nonlinear systems. Both equations are integrable and with remarkable nonlinear waves. His recent work has been devoted to vibration control, and he found two new methods in order to perform this very important task, time-delay state feedback control and nonlocal feedback control.
Foreword
This textbook is devoted to nonlinear physics.
The asymptotic perturbation method is used as a mathematical tool and is explained in some detail, and the theory is developed systematically, starting with nonlinear oscillators, limit cycles and their bifurcations, followed by iterated nonlinear maps, continuous systems, nonlinear partial differential equations (NPDEs), and culminating with infinite-period bifurcation in the nonlinear Schrodinger equation and fractal and chaotic solutions in NPDEs.
A remarkable feature of the book is the emphasis on applications. There are several examples, and the scientific background is explained at an elementary level and closely integrated with the mathematical theory.
This book is ideal for an introductory course at the senior or first-year graduate level. It is also advisable for a scientist who has not a deep knowledge about nonlinear physics but now wants to begin a complete study.
The prerequisites are multivariable calculus and introductory physics.
Introduction
Nonlinear systems are paramount in engineering and science. Many perturbation methods can be used to study these systems in order to predict remarkable bifurcations (a qualitative change in their behavior). In this book, we will use the asymptotic perturbation (AP) method both for nonlinear ordinary differential equations (NODEs) and nonlinear partial differential equations (NPDEs).
In Chapters 1–4, we will study NODEs and will derive a suitable model system to find the most important nonlinear system characteristics. The main finding is that a nonlinear model system of equations describes their behavior. In particular, in Chapter 2, we will describe an infinite-period bifurcation for a parametrically excited Liènard system and find a peculiar attractor for a weakly nonlinear oscillator with a two-period quasiperiodic forcing.
In Chapter 3, we consider vibration control with time-delay state feedback and perform a successful control strategy. In Chapter 4, we illustrate another vibration-control method based on nonlocal dynamics. Numerical simulation confirms our method's validity.
In Chapter 5, we enlarge our perspective and study nonlinear continuous systems, in particular the vibrations of an Euler–Bernoulli beam resting on a nonlinear elastic foundation and with an external periodic excitation. Frequency–response and external force–response curves can be easily found and compared with numerical simulation.
In Chapter 6, the AP method is used to study NPDEs, and we are able to find two new and integrable nonlinear equations, the Maccari system and the Hirota–Maccari equation.
At the same time, the AP method can be used in order to find approximate solutions to relevant physics problems. In Chapter 7, we study the Zakharov–Kusnetsov equation and show the existence of interacting localized solutions because the ZK equation can be described through a C-integrable (solvable via an appropriate change of variables) system of nonlinear evolution equations. Dromions, lumps, ring solitons, and breathers exist for this remarkable nonlinear equation.
In Chapter 8, we study the connection between the AP method and elementary particle physics.
In Chapter 9, we try to explain the rogue waves appearance in nonlinear systems.
In Chapter 10, we arrive at one of the most important findings in this book, fractal and chaotic solutions are possible for nonlinear systems and perhaps at a very fundamental level we must let the particle concept (i.e. a coherent solution) down because we can state that in general solutions have fractal and chaotic properties.
In Chapter 11, we use the AP method in order to arrive at nonlinear quantum mechanics and achieve the Einstein–de Brogle soliton-particle concept by studying the weakly nonlinear Klein–Gordon equation for a particle confined in a box. In Chapter 12, we illustrate how to modify the Einstein equation so as to explain the accelerating and irreversible evolution of the universe. According to Prigogine's ideas, the entropy increase is connected with matter production.
In Chapter 13, this new theory is used to find how confinement and asymptotic freedom can be explained in a framework where particles are like small black holes. Finally, Chapter 14 is devoted to a reverse infinite-period bifurcation for the nonlinear Schrodinger equation in 2 + 1 dimensions.
Many teaching years allowed me writing this book, and I would like to thank my students at Foligno in Perugia University, Italy, for their helpful and valuable suggestions.
1The Asymptotic Perturbation Method for Nonlinear Oscillators
1.1 Introduction
Oscillations are a fundamental topic in physics. When a system is near its equilibrium point, it begins to oscillate, but if the displacement increases, then the nonlinear terms are not negligible. The starting point is the differential equation for the harmonic oscillator
where X(t) is the displacement and ω the circular frequency. The most general solution is
where ρ and θ are fixed by the initial conditions (the Cauchy problem)
if
X
(0) =
X
0
for the displacement
and
for the initial velocity
then we easily get
and
Now, we can consider a weakly nonlinear part in the differential Eq. (1.1) or, on the contrary, a strongly nonlinear part but with small solutions. The first consequence is that the amplitude and the phase are slowly varying with time, so we can introduce another slow time
where ε is a bookkeeping device and q is a rational number that will be chosen afterwards. If we want to study the asymptotic solution behavior (t → ∞) and ε → 0, then τ must assume finite values. So, we assume that an approximate solution is given by
or better
where c. c. stands for complex conjugate and h. o. t. for higher order terms.
Following this path, we are mixing the most important features of two well-known perturbation methods, the harmonic balance and the multiple scale methods (for more details about these two perturbation methods, see Refs. [202, 203, 249]).
If we consider a weakly nonlinear differential equation
where NL stands for the nonlinear part, for instance,
we can insert the solution (1.7) in the nonlinear Eq. (1.8) and with some algebra manipulation, we get for n = 0
then r = 1, for n = 2
and for n = 1
then, q = 2 for the proper nonlinear term balance and with some algebra manipulation
where
We observe that the variable change (1.5) implies that
when the temporal differential operator acts on the function
From Eq. (1.10), we can see that the approximate solution is always periodic, the amplitude is constant, but the period changes and becomes
However, if
the period does not change and is equal to the linear case period.
In this chapter, we want to extend this method and study a generalized Van der Pol–Duffing oscillator in resonance with a periodic excitation
We use the asymptotic perturbation (AP) method based on Fourier expansion and time rescaling (see above) and demonstrate through a second-order perturbation analysis the existence of one or two limit cycles. Moreover, we identify a sufficient condition to obtain a doubly periodic motion when a second low frequency appears, in addition to the forcing frequency. The comparison with the solution obtained by the numerical integration confirms the validity of our analysis.
1.2 Nonlinear Dynamical Systems
The study of nonlinear dynamical systems has interested many researchers, and various methods have been used. Historically, the AP method was first applied in order to study the most important characteristics of a nonlocal oscillator [112, 113, 118].
We now devote our attention to the following type of nonlinear equation
where the dot denotes differentiation with respect to the time and the functions f(x) and g(x, y) are supposed to be analytic.
The limit cycles of the modified Van der Pol equation
have been studied in Ref. [23] by means of a time transformation method.
Phase portraits and dynamical properties of the equation
have been investigated with the methods of differentiable dynamics [74] and the equation
with the method of averaging, the KBM method, the method of multiple scales, and the Poincaré–Lindstedt method [202, 203].
Note that Eqs. (1.22)–(1.24) belong to the general class (1.21) and are characterized by the fact that f(x) is an odd function of x.
We restrict our study to the following particular case of Eq. (1.21)
Eq. (1.5) can be considered a generalized Van der Pol–Duffing equation because it includes as particular cases the Van der Pol oscillator (f2, f3, g1 = 0 and g0 = − g2 ≠ 0) and the Duffing equation (f2 = g1 = g2 = 0 and g0 = f3 ≠ 0). Many authors have studied the problem of approximating the limit cycle of the Van der Pol equation. Stokes [249] used the nonlinear Galerkin method and developed a series representation; Deprit and Schmidt [47] utilized the Poincaré–Lindstedt method to find the amplitude and frequency of the limit cycle; and Garcia-Magallo and Bejarano [57] considered a generalized Van der Pol equation by means of the harmonic balance method. The steady-state behavior of the Van der Pol oscillator has also been studied by integral manifold methods and symbolic manipulation packages by Gilsinn [59, 61]. Mehri and Ghorashi [195] considered the periodically forced Duffing equation in order to establish sufficient conditions to have a periodic solution, and Qaisi [233] studied a similar problem using an analytical approach based on the power series method. In a series of papers [69–71], Hassan used the higher order method of multiple scales with reconstitution and the harmonic balance method to determine the periodic state response of the Duffing oscillator.
In our treatment of Eq. (1.25), no conditions are imposed on the coefficients f2, f3, g1, and g2, which can be of order 1. Only the dissipative coefficient g0 is supposed to be of order e2. Eq. (1.25) transforms into
In the second section, we calculate the approximate solution good to the order of ε4 and construct accurate expressions for the limit cycle of Eq. (1.26). Moreover, we demonstrate that, in the first approximation, the behavior of the solution can be described by means of a model system of differential equations, which represents the characteristics of Eq. (1.26) by means of a reduced set of parameters.
Usually, perturbation analysis is carried out only to the first order because, in many cases, a second order-calculation does not change the qualitative behavior of the solution. However, in Section 1.2, we demonstrate that if the parameters are appropriately chosen, we can find two limit cycles and can calculate their positions only by a second-order perturbation analysis.
In Section 1.3, a comparison with the results of the numerical integration permits discussion of the validity of the AP method.
In Section 1.4, we treat an extension of Eq. (1.26) that is a nonlinear oscillator forced by a small periodic excitation, of order e2, in resonance with the natural frequency of the oscillator
We demonstrate that, under appropriate conditions, a stable limit cycle appears and calculate the relative approximate solution. Moreover, we derive sufficient conditions for the existence of a doubly periodic motion when the fundamental oscillation is subjected to a slight modulation, with an amplitude proportional to the magnitude of the periodic excitation.
Finally, in the last section, we briefly recapitulate the most important results and indicate some possible generalizations of the present study.
1.3 The Approximate Solution
The AP method we use to calculate the approximate solution was first developed in Refs. [1, 2], and then in this section, we sketch the main steps of this perturbation technique.
First of all, we now introduce a rational number
the temporal rescaling
where the rational number q will be fixed afterwards because it establishes to what extent we can push the temporal asymptotic limit in such a way that the nonlinear effects become consistent and not negligible. If t → ∞, then ε → 0, when τ assumes a finite value.
If we take ε = 0 in Eqs. (1.26) and neglect nonlinear terms, we see that it admits simple harmonic solutions X(t) = A exp(−it) + c. c., where A is a constant depending on initial conditions and c. c. stands for complex conjugate. Nonlinear effects induce a modulation of the amplitude A and the appearance of higher harmonics. The modulation is best described in terms of the rescaled variable t that accounts for the need to look on larger time scales, to obtain a nonnegligible contribution from the nonlinear term.
The assumed solution X(t) of (1.26) can be expressed by means of a power series in the expansion parameter ε, we formally write
with γn = |n| for n ≠ 0, and γ0 = r is a positive number, which will be fixed later on; in consequence of the reality of (1.30a)
The assumed solution (1.30a) can be considered a combination of the different harmonics, solutions of the linear equation, i.e. of the equation obtained after neglecting all the nonlinear terms, and the coefficients of this combination depend on τ and ε.
Eq. (1.30a) can be written more explicitly
The functions Ψn(t, ε) depend on the parameter ε, and we suppose that Ψn's limit for ε → 0 exists and is finite and, moreover, they can be expanded in power series of ε, i.e.
In the following, for simplicity, we use the abbreviations for n ≠ 1 and for n = 1.
Note that the variable change (1.29) implies that
After inserting this expansion into Eq. (1.26), we obtain equations for every harmonic and for a fixed order of approximation, which are right for the purpose of determining the coefficients.
For n = 0, we obtain
A correct balance of terms shows r = 2, and then we derive the following relation
For n = 2, taking into account Eq. (1.32), we have
and then
For n = 1, Eq. (1.26) yields for the right-hand side
and for the left-hand side
If q = 2, the first term has the same magnitude order of nonlinear terms.
Taking into account Eqs. (1.33b) and (1.34b), we can derive a differential equation, which involves only Ψ,
with
Substituting the polar form
into Eq. (1.36), and separating real and imaginary parts, we arrive at the following model system:
As we can see from Eqs. (1.30c), (1.31), and (1.40), the approximate solution of Eq. (1.26) can be written as a sum of a contribution of order ε and a contribution of order ε2
By inspection of Eq. (1.41), which can be easily integrated, we conclude that a stable steady-state response is possible if α1 > 0 and β1 < 0. In this case, we obtain a stable equilibrium point, which corresponds to a stable limit cycle for Eq. (1.26), and its approximate expression is given by (1.43), with
The natural frequency of the oscillator is subject to a slight modification and becomes
If we want to improve the validity of the approximate solution, we must include higher order terms. However, we can easily conclude that (for their definition, see Eq. (1.31)). Indeed, we consider Eq. (1.26) for n = 0 and Eqs. (1.33b) and (1.34a) for n = 0 and n = 2 in such a way to obtain
After inserting (1.26b) into (1.26a), we see that the resulting equation is satisfied if . Recall that we can always assume that the initial condition is , because the initial conditions associated with Equation (1.25), X(0) = X0 and , can be used to determine .
A valid higher order approximation can be derived only if we take into account .
For n = 0, we derive the following relation
where h. o. t = higher order terms and
The obvious conclusion is
In a similar way, for n = 2, we obtain
with
If we neglect only terms of order ε6 or higher, Eq. (1.33a–c) transforms into
The term in Eq. (1.50) can be eliminated taking into account that if we differentiate Eq. (1.36), we have
Moreover, from (1.50), we see that it is necessary to consider Eq. (1.26) for n = 3
where
If we use the abbreviation
and Eq. (1.47c), (1.48), and (1.52a), then the final form of Eq. (1.46) is
with
and we arrive at the following modified system model
The approximate solution up to the o(e4)-th order is
where
We can calculate a more accurate expression for the limit cycle, and equations (1.44) and (1.45) become
If we examine the Eq. (1.59) carefully, we can easily understand that there is the possibility of two limit cycles and then of a qualitative change in the behavior of the oscillator with respect to the previsions of the first-order approximation. In fact, we suppose now that the dissipative coefficient g0 is of order e4 and g2 is chosen in such a way that g2 − g1f2 is of order ε, then α1 (see (1.37)) and β1 (see (1.38)) are of order ε4 and ε2, respectively. For example, we can take , g1 = 1, , and f2 = 1.
Taking into account that ρ must be of order e, then all terms in (39) have the same magnitude order and we can obtain for the equilibrium values of ρ two positive roots of order e and then two different limit cycles. Depending on the parameters, the larger limit cycle and the origin can be asymptotically stable and the smaller unstable or vice-versa.
By means of the variable change
which implies
we can always set .
There are four distinct cases:
(i)
: for
α
1
< 0
, there is an unstable limit cycle, and for
α
1
> 0
, no limit cycle appears;
(ii)
: for
α
1
< 0
, there is no limit cycle, and for
α
1
> 0
, only a stable limit cycle appears. Note that cases (i) and (ii) are connected by a simple temporal inversion, followed by the change of the sign of
α
1
.
(iii)
: for
α
1
< 0,
we have a larger stable limit cycle and a smaller unstable limit cycle; for
α
1
> 0
, only the stable limit cycle is present (
Figure 1.1
).
(iv)
: for
α
1
