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Symmetry and perturbation theory in nonlinear dynamics. (English) Zbl 1059.37044

Lecture Notes in Physics. New Series m: Monographs m57. Berlin: Springer (ISBN 3-540-65904-8/hbk). xii, 208 p. (1999).
From the text: This book deals with the theory of Poincaré–Birkhoff normal forms, studying symmetric systems in particular. Attention is focused on general Lie point symmetries, and not just on symmetries acting linearly. Some results on the simultaneous normalization of a vector field describing a dynamical system and vector fields describing its symmetry are presented and a perturbative approach is also used. Attention is given to the problem of convergence of the normalizing transformation in the presence of symmetry, with some other extensions of the theory. The results are discussed for the general case of dynamical systems and also for the specific Hamiltonian setting.
In more detail, symmetry has been a major ingredient in the development of quantum perturbation theory, and it is a basic ingredient of the theory of integrable (Hamiltonian and non-Hamiltonian) systems; yet the use of symmetry in the context of general perturbation theory is rather recent. From the point of view of nonlinear dynamics, the use of symmetry has become widespread only through equivariant bifurcation theory; even in this case, attention has been mostly confined to linear symmetries.
In recent years, the theory and practice of symmetry methods for differential equations has become increasingly popular and has been applied to a variety of problems (following the appearance of the book by P. J. Olver [ Applications of Lie groups to differential equations. Graduate Texts in Mathematics, 107. New York etc.: Springer-Verlag. (1986; Zbl 0588.22001)]). This theory is deeply geometrical and deals with symmetries of general nature (provided that they are described by vector fields), i.e., in this context there is no reason to limit attention to linear symmetries.
“In this book, we look at the basic tools of perturbation theory, i.e., normal forms (first introduced by Poincaré about a century ago) and study their interaction with symmetries, with no limitation to linear ones. We focus on the most basic setting, i.e., systems having a fixed-point (at the origin) and perturbative expansions around this; thus our theory is entirely local.
We have tried to give a reasonably self-contained discussion, so the first three chapters deal with symmetry, differential equations, and in particular dynamical systems in a quite general way (and with many exercises to help the beginner). This part represents a compact introduction to symmetric dynamical systems and does not contain new results.
In Chapters IV and V, we discuss normal forms in the presence of symmetries both in the general (dynamical systems) case and in the Hamiltonian one, at the formal level, i.e., without considering the convergence of the power series entering in the theory. This convergence is studied in Chapter VI, again focusing on the role of symmetries. Many of the results presented in this part are quite recent (mostly between 1994 and 1997), and some of them are actually completely new.
In the final part of the book, we discuss some related problems. In Chapter VII we discuss the relation between symmetries of a dynamical system and its invariant manifolds, in particular the center manifold; in Chapter VIII we discuss an extension of normal forms theory, i.e., the ‘further normalization’ of normal forms; and finally in Chapter IX we discuss a formal approach to asymptotic symmetries.
Obviously this is not a textbook, but we have tried to provide examples and (except in the final chapters) exercises to help the student and/or the beginner in the field.
The discussion given here is rather abstract, although the reader will find some references to applications (of a mathematical and physical nature). On the other hand, in the introduction we have explained in a nontechnical way the general context of the theory discussed and developed in the main body of the book, also giving an extensive list of references dealing with (mostly, very recent) applications of normal forms theory.
The introduction is written in a different style, more similar to that of a short review paper; we hope that this will be useful to the reader who does not already have a strong motivation to study normal forms, without being repulsive to the expert.
Our subject is rather strictly delimited – which allowed us to keep the volume within a reasonable length – and we will not touch upon closely related fields. Together with the length argument, we feel excused for this on the one hand by the premature stage of the development of nonlinear symmetry methods in some of these cognate fields, and on the other by the fact that several good expositions of the available results exist and are mentioned in the introduction.
In particular, we should mention that normal forms are obviously relevant to bifurcation problems, but equivariant bifurcation theory is exhaustively discussed in the book by M. Golubitsky, I. Stewart and D. G. Schaeffer [Singularities and groups in bifurcation theory. Vol. II. Applied Mathematical Sciences, 69. New York etc.: Springer Verlag (1988; Zbl 0691.58003)] (and in the nice short book by G. Iooss and M. Adelmeyer [ Topics in bifurcation theory and applications, Second edition. Advanced Series in Nonlinear Dynamics. 3. Singapore: World Scientific (1998; Zbl 0968.34027)] and in the recent book by P. Chossat and R. Lauterbach [ Methods in equivariant bifurcations and dynamical systems. Advanced Series in Nonlinear Dynamics. 15. Singapore: World Scientific (2000; Zbl 0968.37001)]) in the case of linear symmetries, and not yet so developed in the case of nonlinear symmetries. We hope this book can help in such a development.”

MSC:

37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
34C14 Symmetries, invariants of ordinary differential equations
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
37G05 Normal forms for dynamical systems
37G40 Dynamical aspects of symmetries, equivariant bifurcation theory
37J15 Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010)
70H09 Perturbation theories for problems in Hamiltonian and Lagrangian mechanics
70K60 General perturbation schemes for nonlinear problems in mechanics
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