Toward Analytical Chaos in Nonlinear Systems E-Book

Albert C. J. Luo

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 200

Veröffentlichungsjahr: 2014

Beschreibung

Exact analytical solutions to periodic motions in nonlinear dynamical systems are almost not possible. Since the 18th century, one has extensively used techniques such as perturbation methods to obtain approximate analytical solutions of periodic motions in nonlinear systems. However, the perturbation methods cannot provide the enough accuracy of analytical solutions of periodic motions in nonlinear dynamical systems. So the bifurcation trees of periodic motions to chaos cannot be achieved analytically. The author has developed an analytical technique that is more effective to achieve periodic motions and corresponding bifurcation trees to chaos analytically. Toward Analytical Chaos in Nonlinear Systems systematically presents a new approach to analytically determine periodic flows to chaos or quasi-periodic flows in nonlinear dynamical systems with/without time-delay. It covers the mathematical theory and includes two examples of nonlinear systems with/without time-delay in engineering and physics. From the analytical solutions, the routes from periodic motions to chaos are developed analytically rather than the incomplete numerical routes to chaos. The analytical techniques presented will provide a better understanding of regularity and complexity of periodic motions and chaos in nonlinear dynamical systems. Key features: * Presents the mathematical theory of analytical solutions of periodic flows to chaos or quasieriodic flows in nonlinear dynamical systems * Covers nonlinear dynamical systems and nonlinear vibration systems * Presents accurate, analytical solutions of stable and unstable periodic flows for popular nonlinear systems * Includes two complete sample systems * Discusses time-delayed, nonlinear systems and time-delayed, nonlinear vibrational systems * Includes real world examples Toward Analytical Chaos in Nonlinear Systems is a comprehensive reference for researchers and practitioners across engineering, mathematics and physics disciplines, and is also a useful source of information for graduate and senior undergraduate students in these areas.

Leseprobe

Table of Contents

Cover

Title Page

Copyright

Preface

Chapter 1: Introduction

1.1 Brief History

1.2 Book Layout

Chapter 2: Nonlinear Dynamical Systems

2.1 Continuous Systems

2.2 Equilibriums and Stability

2.3 Bifurcation and Stability Switching

Chapter 3: An Analytical Method for Periodic Flows

3.1 Nonlinear Dynamical Systems

3.2 Nonlinear Vibration Systems

3.3 Time-Delayed Nonlinear Systems

3.4 Time-Delayed, Nonlinear Vibration Systems

Chapter 4: Analytical Periodic to Quasi-Periodic Flows

4.1 Nonlinear Dynamical Systems

4.2 Nonlinear Vibration Systems

4.3 Time-Delayed Nonlinear Systems

4.4 Time-Delayed, Nonlinear Vibration Systems

Chapter 5: Quadratic Nonlinear Oscillators

5.1 Period-1 Motions

5.2 Period-

m

Motions

5.3 Arbitrary Periodical Forcing

Chapter 6: Time-Delayed Nonlinear Oscillators

6.1 Analytical Solutions

6.2 Analytical Bifurcation Trees

6.3 Illustrations of Periodic Motions

References

Index

End User License Agreement

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Guide

Cover

Table of Contents

Preface

Chapter 1: Introduction

List of Illustrations

Figure 2.1

Figure 5.1

Figure 5.2

Figure 5.3

Figure 5.4

Figure 5.5

Figure 5.6

Figure 5.7

Figure 5.8

Figure 5.9

Figure 5.10

Figure 5.11

Figure 5.12

Figure 5.13

Figure 5.14

Figure 5.15

Figure 6.1

Figure 6.2

Figure 6.3

Figure 6.4

Figure 6.5

List of Tables

Table 5.1

Table 5.2

Table 6.1

TOWARD ANALYTICAL CHAOS IN NONLINEAR SYSTEMS

Albert C. J. Luo

Southern Illinois University, USA

This edition first published 2014

© 2014 John Wiley & Sons Ltd

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Library of Congress Cataloging-in-Publication Data

Luo, Albert C. J.

Toward analytical chaos in nonlinear systems / Albert C. J. Luo.

pages cm

Includes bibliographical references and index.

ISBN 978-1-118-65861-1 (hardback)

1. Differentiable dynamical systems. 2. Nonlinear oscillations. 3. Chaotic behavior in systems. I. Title.

QA867.5.L86 2014

003′.857—dc23

2014001972

A catalogue record for this book is available from the British Library.

Preface

There has been interest in periodic solutions of dynamical systems for a few centuries. The periodic motion as a steady-state motion has attracted many scientists' attention. Until now, it still cannot be achieved analytically and accurately. In the second half of the twentieth century, using computer software, another steady-state motion (i.e., chaos) has been observed to be caused by bifurcations of periodic motions. Such a steady-state motion can be numerically simulated with computational errors. Other than that, it was not known how to obtain the bifurcation trees of periodic motions to chaos analytically in nonlinear dynamical systems, and the mathematical structures of solutions for chaos are still unknown. This is because it is not yet known how to obtain the exact solutions of periodic motions analytically. This book will try to solve this problem. Further, the title of this book is Toward Analytical Chaos in Nonlinear Systems. The author hopes this book can attract more attention to finding accurate analytical solutions of periodic motions to chaos, and the mathematical structures of chaos solutions can be achieved.

Since 1788, Lagrange has used the fundamental matrix of the linearized system as a moving coordinate transformation to obtain the Lagrange stand form, which is based on the variation of parameter procedure. From the Lagrange standard form, the method of averaging was used to investigate the gravitational three-body problem through a two-body problem with a perturbation. At the end of the nineteenth century, Poincare developed the perturbation method to investigate the periodic motion of the three-body problem. Since then, this cycle of perturbation method for periodic motions has remained, and it is hard to change such a cycle. Indeed, the perturbation method helps one understand some nonlinear phenomena and has given some reasonable explanations. However, based on the Lagrange standard form or perturbation methods, vector fields in dynamical systems have been changed, and the deformed vector fields cannot represent the original nonlinear dynamical systems. From the idea of the Lagrange standard form, the normal forms of nonlinear dynamical systems at a state of equilibrium cannot be used for periodic motions and chaos in the original nonlinear dynamical systems.

This book provides an analytical method for determining periodic flows and quasi-periodic flows in nonlinear dynamical systems with/without time-delay. From the analytical solutions of periodic motions, the bifurcation trees of periodic flows to chaos can be determined analytically. Further, one can achieve analytical solutions of chaos and understand the corresponding mathematical structures. The method presented in this book gives frequency-responses for nonlinear dynamical systems as the Laplace transformation for linear dynamical systems. This book has six chapters. Chapter 1 gives a brief history of the study of periodic motions in nonlinear dynamical systems. The stability, stability switching, and bifurcation of equilibriums in nonlinear continuous systems are introduced in Chapter 2, which is different from the traditional presentation. In Chapter 3, the analytical method for period-m flows in dynamical systems is presented, including autonomous and non-autonomous dynamical systems with/without time delay. In Chapter 4, the analytical determination of period-m to quasi-periodic flows is presented. In Chapter 5, the analytical bifurcation tree of periodic motion to chaos is presented through a periodically excited, quadratic nonlinear oscillator. In Chapter 6, the analytical solution for a time-delayed, quadratic nonlinear system is presented. The materials presented in this book will provide a different way to achieve analytical solutions of periodic flows to chaos in nonlinear dynamical systems.

Finally, I would like to thank my students (Bo Yu and Hanxiang Jin) for applying the recently developed analytical method to two nonlinear systems and completing numerical computations. Also, I would like to thank my wife (Sherry X. Huang) and my children (Yanyi Luo, Robin Ruo-Bing Luo, and Robert Zong-Yuan Luo) again for tolerance, patience, understanding, and continuous support.

Albert C.J. LuoEdwardsville, Illinois, USA

Chapter 1Introduction

In this chapter, a brief literature survey of analytical solutions of periodic motions in nonlinear dynamical systems will be presented. The perturbation analysis has played an important role in such an approximate analysis of periodic motions in nonlinear systems. The perturbation method, method of averaging, harmonic balance, and generalized harmonic balance will be reviewed. The application of perturbation method in time-delayed systems will be discussed briefly.

1.1 Brief History

Since the seventeenth century, there has been interest in periodic motions in dynamical systems. The Fourier series theory shows that any periodic function can be expressed by a Fourier series expansion with different harmonics. In addition to simple oscillations, there has been interest in the motions of moon, earth, and sun in the three-body problem. The earliest approximation method is the method of averaging, and the idea of averaging originates from Lagrange (1788). At the end of the nineteenth century, Poincare (1890) provided the qualitative analysis of dynamical systems to determine periodic solutions and stability, and developed the perturbation theory for periodic solutions. In addition, Poincare (1899) discovered that the motion of a nonlinear coupled oscillator is sensitive to the initial condition, and qualitatively stated that the motion in the vicinity of unstable fixed points of nonlinear oscillation systems may be stochastic under regular applied forces. In the twentieth century, one followed Poincare's ideas to develop and apply the qualitative theory to investigate the complexity of motions in dynamical systems. With Poincare's influence, Birkhoff (1913) continued Poincare's work, and proof of Poincare's geometric theorem was given. Birkhoff (1927) showed that both stable and unstable fixed points of nonlinear oscillation systems with two degrees of freedom must exist whenever their frequency ratio (or called resonance) is rational. The sub-resonances in periodic motions of such systems change the topological structures of phase trajectories, and the island chains are obtained when the dynamical systems can be renormalized with fine scales. In such qualitative and quantitative analysis, the Taylor series expansion and the perturbation analysis play an important role. However, the Taylor series expansion analysis is valid in the small finite domain under certain convergent conditions, and the perturbation analysis based on the small parameters, as an approximate estimate, is only acceptable for a very small domain with a short time period.

van der Pol (1920) used the averaging method to determine the periodic motions of self-excited systems in circuits, and the presence of natural entrainment frequencies in such a system was observed in van der Pol and van der Mark (1927). Cartwright and Littlewood (1945) discussed the periodic motions of the van der Pol equation and proved the existence of periodic motions. Cartwright and Littlewood (1947) discussed the periodic motions of a generalized nonlinear equation based on the similar Duffing equation. Levinson (1948) used a piecewise linear model to describe the van der Pol equation and determined the existence of periodic motions. Levinson (1949) further developed the structures of periodic solutions in such a second order differential equation through the piecewise linear model, and discovered that infinite periodic solutions exist in such a piecewise linear model. From the Levinson's results, Smale (1967) used the topological point to present the Smale horseshoe with discontinuous mappings to describe the existence of infinite periodic motions. Further, a differentiable dynamical system theory was developed. Such a theory has been extensively used to interpret the homoclinic tangle phenomenon in nonlinear dynamical systems. Smale found the infinite, many periodic motions, and a perfect minimal Cantor set near a homoclinic motion can be formed. Melnikov (1962) used the concept of Poincare (1892) to investigate the behavior of trajectories of perturbed systems near autonomous Hamiltonian systems. Melnikov (1963) further investigated the behavior of trajectories of perturbed Hamiltonian systems, and the width of the separatrix splitting was approximately estimated. The width gives the domain of the chaotic motion in the vicinity of the generic separatrix. Even if the width of the separatrix splitting was approximately estimated, the dynamics of the separatrix splitting was not developed.

Since the nonlinear phenomena was observed in engineering, Duffing (1918) used the hardening spring model to investigate the vibration of electro-magnetized vibrating beam, and after that, the Duffing oscillator has been extensively used in structural dynamics. In addition to determining the existence of periodic motions in nonlinear differential equations of the second order in mathematics, one has applied the Poincare perturbation methods for periodic motions in nonlinear dynamical systems. Fatou (1928) provided the first proof of asymptotic validity of the method of averaging through the existence of solutions of differential equations. Krylov and Bogolyubov (1935) systematically developed the method of averaging and the detailed discussion can be found in Bogoliubov and Mitropolsky (1961). The classic perturbation methods for nonlinear oscillators were presented (e.g., Stoker, 1950; Minorsky, 1962; Hayashi, 1964). Hayashi (1964) used the method of averaging and harmonic balance method to discuss the approximate periodic solutions of nonlinear systems and the corresponding stability. Nayfeh (1973) employed the multiple-scale perturbation method to develop approximate solutions of periodic motions in the Duffing oscillators. Holmes and Rand (1976) discussed the stability and bifurcation of periodic motions in the Duffing oscillator. Nayfeh and Mook (1979) applied the perturbation analysis to nonlinear structural vibrations via the Duffing oscillators, and Holmes (1979) demonstrated chaotic motions in nonlinear oscillators through the Duffing oscillator with a twin-well potential. Ueda (1980) numerically simulated chaos via period-doubling of periodic motions of Duffing oscillators.

Based on the work of Melnikov (1963), Greenspan (1981) extended the similar ideas to the dissipative dynamical systems (also see, Greenspan and Holmes, 1983; Guckenheimer and Holmes, 1983). Further, the Melnikov method was developed for the global transversality in dissipative nonlinear systems. Once the global transversality to the separatrix exists, one thought that the Smale horseshoe presented in Smale (1967) may exist, and furthermore chaos in such a nonlinear dynamical system may occur. However, from such a prediction based on the Melnikov method, one cannot observe the global transversality in nonlinear dynamical systems. The Smale horseshoe theory may not be adequate for nonlinear dynamical systems rather than the topological structure. From the perturbation analysis, the Melnikov function was obtained for Hamiltonian systems with a small perturbation. One used such a function to analytically predict global behaviors (e.g., chaos) in the Hamiltonian systems with a small perturbation. Because of the perturbation analysis, the Melnikov method can give a reasonable analysis of the global behavior only when the perturbation is very small and close to zero. However, the perturbation is very small to zero, chaos in nonlinear dynamical systems may not occur. So the Melnikov method may not help us understand the global behaviors of nonlinear dynamical systems. Luo (1995) used the Chirikov criterion to determine Hamiltonian chaos and applied the Melnikov function to investigate the global transversality (also see, Luo and Han, 1999; Luo, 2008 2012a). The conclusion is that the Melnikov method cannot provide an adequate prediction of chaotic motions in the dissipative system. For a better understanding of the Melnikov method, the work of Melnikov (1963) should be revisited. Melnikov (1963) presented a perturbation analysis to estimate the width of the separatrix splitting. Indeed, the width of the separatrix can be approximately estimated, but it cannot be used for predicting the existence of chaos. The Melnikov function is an approximate energy increment during a certain time period, which can be found in references (e.g., Arnold, 1964; Chirikov, 1979; Luo and Han, 2001). If the Melnikov function is zero, from a physical point of view, the system energy is conserved during a certain time period. Such a zero value of the Melnikov function does not imply that the flow has any global transversality to the separatrix. One has difficulty finding a connection from periodic motions to chaos. Thus, one continues using the perturbation analysis to determine the approximate analytical solutions of periodic motions. Coppola and Rand (1990) determined limit cycles of nonlinear oscillators through elliptic functions in the averaging method. Wang et al. (1992) used the harmonic balance method and the Floquet theory to investigate the nonlinear behaviors of the Duffing oscillator with a bounded potential well (also see, Kao et al., 1992). Luo and Han (1997) determined the stability and bifurcation conditions of periodic motions of the Duffing oscillator. However, only symmetric periodic motions of the Duffing oscillators were investigated. Luo and Han (1999) investigated the analytical prediction of chaos in nonlinear rods through the Duffing oscillator. Peng et al. (2008) presented the approximate symmetric solution of period-1 motions in the Duffing oscillator by the harmonic balance method with three harmonic terms. Luo (2012a) developed a generalized harmonic balance method to get the approximate analytical solutions of periodic motions and chaos in nonlinear dynamical systems. This method used the finite Fourier series to express periodic motions and the coefficients are time-varying. With averaging, a dynamical system of coefficients are obtained from which the steady-state solution are achieved and the corresponding stability and bifurcation are completed. Luo and Huang (2012a) used the generalized harmonic balance method with finite terms to obtain the analytical solution of period-1 motion of the Duffing oscillator with a twin-well potential. Luo and Huang (2012b) employed a generalized harmonic balance method to find analytical solutions of period-m motions in such a Duffing oscillator. The analytical bifurcation trees of periodic motions in the Duffing oscillator to chaos were obtained (also see, Luo and Huang, 2012c,d, 2013a,b,c, 2014). Such analytical bifurcation trees show the connection from periodic solution to chaos analytically. To better understand nonlinear behaviors in nonlinear dynamical systems, the analytical solutions for the bifurcation trees from period-1 motion to chaos in a periodically forced oscillator with quadratic nonlinearity were presented in Luo and Yu (2013a, b, c), and period-m motions in the periodically forced, van der Pol equation was presented in Luo and Laken (2013). The analytical solutions for the van der Pol oscillator can be used to verify the conclusions in Cartwright and Littlewood (1947) and Levinson (1949). The results for the quadratic nonlinear oscillator in Luo and Yu (2013a, b, c) analytically show the complicated period-1 motions and the corresponding bifurcation structures.

In recent years, time-delayed systems are of great interest since such systems extensively exist in engineering (e.g., Tlusty, 2000; Hu and Wang, 2002). The infinite dimensional state space causes a significant difficulty in understanding such a time-delayed problem. One tried to work on numerical methods to get the corresponding complicated behaviors. On the other hand, one is interested in the stability and bifurcation of equilibriums of the time-delayed systems (e.g., Stepan, 1989; Sun, 2009; Insperger and Stepan, 2011). In addition, one is also interested in periodic solutions in time-delayed dynamical systems. Perturbation methods have been used in recent years for such periodic motions in delayed dynamical systems. For instance, the approximate solutions of the time-delayed nonlinear oscillator were investigated by the method of multiple scales (e.g., Hu, Dowell, and Virgin 1998; Wang and Hu, 2006). The harmonic balance method was also used to determine approximate periodic solutions for delayed nonlinear oscillators (e.g., MacDonald, 1995; Liu and Kalmar-Nagy, 2010; Leung and Guo, 2012). However, such approximate solutions of periodic motions in the time-delayed oscillators are based on one or two harmonic terms, which are not accurate enough. In addition, the corresponding stability and bifurcation analysis of such approximate solutions may not be adequate. In this book, an alternative way of finding the accurate analytical solutions of periodic flows in time-delayed dynamical systems will be presented. This method is without any small-parameter requirement. In addition, this approach can also be applicable to the coefficient varying with time.

1.2 Book Layout

In this book, a new analytical method will be presented for analytical solutions of periodic motions in nonlinear dynamical systems with/ without time delay. The basic theory of nonlinear systems will be briefly introduced. The analytic method based on the generalized harmonic balance will be comprehensively discussed, and this method will be applied to nonlinear dynamical systems to find the periodic motions analytically and to determine the analytical bifurcation trees of periodic motions to chaos. The main body in this book is summarized as follows:

In Chapter 2, the basic theory of nonlinear dynamical systems will be introduced. Local theory, global theory, and bifurcation theory of nonlinear dynamical systems will be briefly discussed. The stability switching and bifurcation on specific eigenvectors of the linearized system of a nonlinear system at equilibrium will be discussed. The higher-order singularity and stability for nonlinear systems on the specific eigenvectors will be developed.

In Chapter 3, from Luo (2012a), the analytical dynamics of periodic flows and chaos in nonlinear dynamical systems will be presented. The analytical solutions of periodic flows and chaos in autonomous systems will be discussed first, and the analytical dynamics of periodically forced nonlinear dynamical systems will be presented. The analytical solutions of periodic motions in free and periodically forced vibration systems will be presented. In a similar fashion, the analytical solutions of periodic flows for time-delayed nonlinear systems will be presented with/without periodic excitations, and time-delayed nonlinear vibration systems will be also discussed for engineering application. The analytical solutions of periodic flows and chaos are independent of the small parameters, which are different from the traditional perturbation methods. The methodology presented herein will end the history of chaos being numerically simulated only.

In Chapter 4, from the idea of Luo (2012a, 2013), period-

m

flows to quasi-periodic flows in nonlinear dynamical systems will be presented. The analytical solutions of quasi-periodic flows in autonomous systems will be discussed, and the analytical solutions of quasi-periodic flows in periodically forced nonlinear dynamical systems will be presented. The analytical solutions of quasi-periodic motions in free and periodically forced vibration systems will be presented. The analytical solutions of quasi-periodic flows for time-delayed nonlinear systems will be presented with/without periodic excitations, and time-delayed nonlinear vibration systems will be discussed as well.

In Chapter 5, analytical solutions for period-

m

motions in a periodically forced, quadratic nonlinear oscillator will be presented through the Fourier series solutions with finite harmonic terms, and the stability and bifurcation analyses of the corresponding period-1 motions will be carried out. There are many period-1 motions in such a nonlinear oscillator, and the parameter map for excitation amplitude and frequency will be developed for different period-1 motions. For each period-1 motion branch, analytical bifurcation trees of period-1 motions to chaos will be presented. For a better understanding of complex period-

m

motions in such a quadratic nonlinear oscillator, trajectories, and amplitude spectrums will be illustrated numerically.

In Chapter 6, analytical solutions for period-

m

motions in a time-delayed, nonlinear oscillator will be presented through the Fourier series, and the stability and bifurcation analyses of the corresponding periodic motions will be presented through the eigenvalue analysis. Analytical bifurcation trees of periodic motions to chaos will be presented through the frequency-amplitude curves. Trajectories and amplitude spectrums of periodic motions in such a time-delayed nonlinear system will be illustrated numerically for a better understanding of time-delayed nonlinear dynamical systems.

Chapter 2Nonlinear Dynamical Systems

In this chapter, the basic theory of nonlinear dynamical systems will be introduced. Local theory, global theory, and bifurcation theory of nonlinear dynamical systems will be briefly discussed. The stability switching and bifurcation on specific eigenvectors of the linearized system at equilibrium will be discussed. The higher-order singularity and stability for nonlinear systems on the specific eigenvectors will be developed.

2.1 Continuous Systems

Definition 2.1

For , , and , consider a vector function which is ()-continuous, and there is an ordinary differential equation in a form of

2.1

where is derivative with respect to time , which is simply called the velocity vector of the state variables . With an initial condition of , the solution of Equation (2.1) is given by

2.2

The ordinary differential equation with the initial condition is called a dynamical system.

The vector function

is called a vector field on domain

.

The solution

is called the flow of dynamical system.

The projection of the solution

on domain

is called the trajectory, phase curve, or orbit of dynamical system, which is defined as

2.3

Definition 2.2

If the vector field of the dynamical system in Equation (2.1) is independent of time, such a system is called an autonomous dynamical system. Thus, Equation (2.1) becomes

2.4

Otherwise, such a system is called non-autonomous dynamical systems if the vector field of the dynamical system in Equation (2.1) is dependent on time and state variables.

Definition 2.3

For a vector function with , the operator norm of is defined by

2.5

For with an matrix , the corresponding norm is defined by

2.6

Definition 2.4

For a vector function the derivative and integral of are defined by

2.7

For an matrix , the corresponding derivative and integral are defined by

2.8

Definition 2.5

For , , and , the vector function with is differentiable at if

2.9

is called the spatial derivative of at