Nonlinear Dynamics: Exploration Through Normal Forms

Description

Geared toward advanced undergraduates and graduate students, this exposition covers the method of normal forms and its application to ordinary differential equations through perturbation analysis. In addition to its emphasis on the freedom inherent in the normal form expansion, the text features numerous examples of equations, the kind of which are encountered in many areas of science and engineering. The treatment begins with an introduction to the basic concepts underlying the normal forms. Coverage then shifts to an investigation of systems with one degree of freedom that model oscillations, in which the force has a dominant linear term and a small nonlinear one. The text considers a variety of nonautonomous systems that arise during the study of forced oscillatory motion. Topics include boundary value problems, connections to the method of the center manifold, linear and nonlinear Mathieu equations, pendula, Nuclear Magnetic Resonance, coupled oscillator systems, and other subjects. 1998 edition.

The book under review is declared to be the first one devoted exclusively to exploring the many possibilities afforded by normal forms as applied to nonlinear systems modeled by ordinary differential equations that are amenable to perturbative analysis. The authors develop a detailed and self-contained exposition of this investigative tool featuring numerous simple examples, in particular, ones arising in various areas of physics. Therefore, this book can serve as a professional reference for engineers and a course text for advanced-level students in nonlinear dynamics in physics, applied mechanics, and applied mathematics. Weakly nonlinear systems of the form x˙=Ax+εF(ε,x)\dot {\bold x}=\bold{Ax}+\varepsilon{\bold F}(\varepsilon,{\bold x}) are primarily considered where ε\varepsilon is a small parameter, x{\bold x} is a multidimensional vector, and, for simplicity of the algebra, the constant matrix A{\bold A} is often assumed to be diagonal. Two-dimensional systems, in particular, near-conservative oscillators with one degree of freedom are especially examined.
Chapter 1 introduces basic ideas associated with perturbation methods as applied to problems in nonlinear dynamics, particularly the idea of a near-identity transformation and the role it plays in the normal form expansion. Two basic simple models (the conservative Duffing oscillator and the van der Pol oscillator) are also introduced that will be used in the subsequent chapters to illustrate different aspects of the method of normal forms. Here and throughout the book the authors emphasize the freedom or nonuniqueness inherent to the normal form expansion. This contrasts to naive perturbation theory where only the solutions of the unperturbed problem are used as generators of a series expansion for the solutions of the full problem. Chapter 2 introduces some basic concepts in the theory of dynamical systems such as usual definitions and fundamental theorems without proofs as well as Gronwall's lemma. Chapter 3 discusses the naive perturbation theory and shows that it is successful in some cases, but for one-degree-of-freedom conservative oscillator systems it leads to spurious secular terms. In Chapter 4, the formalism of the perturbation expansion associated with the method of normal forms is developed. The basic idea by Poincaré allowing one to avoid the secular terms is to seek the zero-order approximation of the solutions of the full system as to obey an equation including ε\varepsilon-dependent terms. The concepts of resonance, Poincaré and Siegel domains, and convergence of normal form transformation are briefly discussed. Simple planar systems (primarily, the Duffing oscillator) are studied to illustrate various aspects of the normal form expansion and to show explicitly how one computes various terms. Finally, the validity of the truncated expansions for planar near-conservative oscillatory systems is studied.
In Chapter 5, problems are discussed in which the eigenvalues of the unperturbed system have negative real parts. For such problems, the naive perturbation theory yields a perturbation expansion that is equivalent to the normal form expansion due to the fact that only a finite number of resonant terms are possible and, as a result, the generators of the expansion can be taken to be the unperturbed solutions. Some aspects of finite-time blowup are also discussed. In this chapter, one-dimensional systems with quadratic nonlinearity are primarily used to illustrate the results. However, numerical simulations of two- and three-dimensional systems are also presented that are related to the blowup phenomena. In Chapters 6 and 7, the authors study, respectively, conservative and near-conservative damped oscillatory systems with one degree of freedom. Here, the normal form analysis provides a faithful perturbation description of the motion, in contrast to the naive perturbation theory. For the Duffing equation, different choices of free functions are discussed and the equivalence of all the possible choices is shown. The concepts of the so-called minimal normal form and limit circle radius renormalization are introduced and treated. Other topics under consideration are precision and duration of the validity of the expansion and bifurcations in a boundary-value problem.
Chapter 8 contains the study of a rich variety of nonautonomous time-periodic systems in resonance or near-resonance regimes. All these systems are near-linear, with two phase variables; most of them arise in physics. Discussing the nonlinear Mathieu equation, the authors expose the spurious "explosive instability''. This means that the solutions of the first-order approximation exhibit a finite-time blowup, while the solutions of the full problem are well behaved and well described by the solutions of the second approximation. Chapter 9 is devoted to a discussion of techniques for the treatment of problems in which the unperturbed part has one or more zero eigenvalues and the remaining ones have negative real parts. The method of normal forms, without modification, is found to provide a faithful characterization of such systems. In contrast, the wide-spread method of center manifold implies the restriction to both sufficiently small initial amplitudes and a neighbourhood of the origin. It cannot also follow the transient behaviour. The connection is shown between the results obtained by both the methods for two-dimensional systems with one zero eigenvalue and one negative eigenvalue. In particular, the equation derived in a center manifold perturbation analysis is precisely the zero-order equation for the zero-eigenvalue component within the method of normal forms.
In Chapter 10, Hamiltonian systems consisting of several harmonic oscillators with a small nonlinear coupling are examined. The case of two oscillators is considered in more detail. The normal form expansion for a planar near-rotation symplectic map is also discussed. Chapter 11 is devoted to some phenomena in higher-dimensional dissipative systems that are amenable to the perturbative analysis. These include the presence or onset of a limit cycle as an asymptotic solution of two leading oscillatory coordinates, in particular, Hopf bifurcations, and the phenomenon of phase locking in a system of two coupled oscillators. The first phenomenon can be analyzed by the method of the center manifold, provided that the eigenvalues for the nonleading coordinates have negative real parts (those for the leading coordinates are pure near-imaginary). Nevertheless, an appropriate three-dimensional example is discussed where the eigenvalue of a single nonleading coordinate is close to zero. The appendix contains some technical details concerning the calculation of the period of oscillations of a one-degree-of-freedom oscillator in a perturbed quadratic potential.

Reviewed by Sergei A. Dovbysh