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Methods of qualitative theory in nonlinear dynamics. Part I. (English) Zbl 0941.34001

World Scientific Series on Nonlinear Science. Series A. 4. Singapore: World Scientific. xxiv, 392 p. (1998).
The study of structural stability and equivalent systems of differential equations is one of the most exciting topics in the theory of differential equations. This book gives the reader a taste for the qualitative theory of ordinary differential equations by providing an accessible exposition of the character of structurally stable equilibrium states as well as of structurally stable periodic trajectories of dynamical systems. It is based on related courses, which the first author, L. P. Shilnikov, gave at Gorky University during the last thirty years.
The book contains six chapters and two appendices. Chapter 1 is introductory and it describes the principal properties of autonomous systems, when they are faced as dynamical systems. Some facts from the basic theory of abstract dynamical systems (as Poisson stability) are also presented, where a brief reference is done to the notion of the topological equivalence.
Chapter 2 provides an analysis of the behavior of trajectories in a neighborhood of a structurally stable equilibrium state. Attention is given on the behavior of the solutions near a saddle point, as well as on the invariant stable and unstable manifolds at such a point. In the last section of the chapter, the problem of smooth linearization is discussed, where some useful information on the Poincaré theory of resonances for local bifurcation is given.
In Chapter 3 the authors discuss structurally stable periodic trajectories and give a complete picture of the behavior of the Poincaré mapping near a fixed point. The two-dimensional case is presented first and then in the high-dimensional case by using topological conjugacy they provide a classification of systems of differential equations near a structurally stable periodic trajectory. The case of saddle fixed points is discussed in detail, where a theorem on the existence of invariant manifolds for such points is proved.
Invariant tori under periodic or quasi-periodic nonautonomous systems are considered in Chapter 4. After discussing the existence of an invariant torus the book contains a theorem on the persistence of an invariant torus for a relatively small periodic smooth perturbation of an autonomous system, when the later has a stable periodic trajectory. In the case of a periodic external force, the behavior of the trajectories on a two dimensional invariant torus may be modeled by an orientable diffeomorphism of a circle. The chapter closes with a discussion of the synchronization problem (the van der Pol equation) associated with the phenomenon of beats in modulations.
Chapter 5 presents a consideration of local manifolds, namely the fact that in a small neighborhood of a structurally unstable equilibrium there is locally an invariant smooth center manifold, whose dimension is equal to the number of the characteristic exponents with zero real part. The proof of these results, which is based on a boundary value problem, is given in the last section.
Results on periodic solutions similar to those of Chapter 5 are presented in Chapter 6. This chapter contains the center manifold theorem for a homoclinic loop. The formulation of the center manifold theorems is given in Section 1, while by using the Poincaré mapping near such a loop, the proof is presented in Section 6.3. The chapter closes with the examination of the form of the center manifold theorem for heteroclinic cycles.
In Appendix A a theorem is given on the reduction of a system to a special form which is suitable for the analysis of the trajectories near a saddle point. The importance of such a reduction over a simple linearization near a saddle point is obvious, even for local results. Finally in Appendix B the case of first-order asymptotics for the trajectories near a saddle fixed point is investigated.
The authors have adapted a free style of presentation that avoids the more common definition-theorem-proof style used in most mathematical books. The readers may appreciate the book’s concrete approach which centers on results rather than theorems. Another characteristic which many readers may appreciate is that some theorems with rather long proof are firstly investigated by discussing particular cases and finally their proof is presented. For instance, the proof of theorems 2.5, 2.6, 2.8, 3.4, 3.6 are completed or given in Chapter 5, theorem 2.7 is proved in Section 2.8, theorem 2.17 in Appendix A, Lemma 3.6 in Appendix B, etc. On the other hand some formal definitions of important terms are not always given. For example, the structural stability is defined on page 24 via the nature of the characteristic exponents, but the definition of the notion is not given for the general not necessary smooth case. The book contains 81 references.
This book can be used at an introductory graduate level and also it will be useful to any reader interested in the qualitative behavior of solutions to smooth differential equations close to simple equilibrium states and periodic trajectories, as well as in the principal bifurcations of equilibrium states or periodic homoclinic and heteroclinic trajectories.

MSC:

34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
34C41 Equivalence and asymptotic equivalence of ordinary differential equations
34D30 Structural stability and analogous concepts of solutions to ordinary differential equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
34C30 Manifolds of solutions of ODE (MSC2000)
34C25 Periodic solutions to ordinary differential equations
37J25 Stability problems for finite-dimensional Hamiltonian and Lagrangian systems
37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems
37B25 Stability of topological dynamical systems