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Multifrequency nonlinear oscillations and their bifurcations. (Mnogochastotnye nelinejnye kolebaniya i ikh bifurkatsii.) (Russian) Zbl 0791.34032
Leningrad: Leningradskij Gosudarstvennyj Universitet. 144 p. (1991).
The book is devoted to invariant tori and quasiperiodic motions in multidimensional nonlinear dynamical systems. It consists of the introduction, three chapters and three appendices. In the introduction, the author gives a motivation for the subject and states some auxiliary results. He recalls that the search for periodic solutions, or closed trajectories, is one of the main topics in the theory of ODEs. There are two natural multidimensional generalizations of a periodic solution, namely, a quasiperiodic solution and an invariant torus. In Chapter I, the author considers systems of the form \(dx/dt=X (x, \varepsilon)\) with \(x \in \mathbb{R}^ d\), \(\varepsilon \in \mathbb{R}\), \(X\) sufficiently smooth \((X(0,0) =0\), the matrix \(D_ xX(0,0)\) possessing purely imaginary eigenvalues), and describes an interesting and powerful procedure, due to himself, for constructing invariant tori of various dimensions which bifurcate from the origin as the parameter \(\varepsilon\) passes through the critical value 0. The procedure is iterative and consists of a finite number of steps. The main ingredient of the proof is a certain contracting integral operator. Some generalizations are also presented, in particular, the case of a multidimensional parameter \(\varepsilon \in \mathbb{R}^ m\) and a bifurcation of an invariant torus from an invariant torus of smaller dimension. In Chapter II, the author develops a more classical theme about quasiperiodic solutions of analytic autonomous (or time-periodic) ODEs. First he proves, via a KAM-like infinite sequence of coordinate transformations, a powerful general theorem on the existence of quasiperiodic solutions. This theorem is then applied to various classes of systems. The author emphasizes the fundamental difference in the behavior of quasiperiodic motions in systems of generic type and those without dissipation (in particular, Hamiltonian and reversible systems). For generic systems, one should let an equation depend on a multidimensional parameter, and then the system is expected to admit an isolated quasiperiodic solution over a Cantor set in the parameter space. On the other hand, an individual dissipation-free (or neutral, in a sense to be made precise) system, without any parameters, usually has Cantor families on invariant tori which carry quasiperiodic motions. Quasiperiodic solutions of Hamiltonian and reversible systems are studied in detail in various contexts, including lower-dimensional tori (but only when the additional variables exhibit hyperbolicity). The author emphasizes the full similarity in the properties of these two classes of neutral systems. In Chapter III, the results of the first two chapters are specialized to two-dimensional systems periodic in time. In this case the existence of many invariant tori often implies Lyapunov stability. Again, much attention is paid to Hamiltonian and reversible systems. The first two appendices, written by the author’s Ph. D. candidates V. V. Basov and S. P. Tokarev, are devoted to normalizing transformations of two-dimensional autonomous systems in the case of nonrough focus (i.e., focus with the linear terms of the center type.) Two theorems are proven, that the formal normalizing transformation of an analytic system is divergent if the coefficients of the Taylor series satisfy certain inequalitities, and that a \(C^ \infty\) system and its \(C^ \infty\) normal form are conjugated by a \(C^ \infty\) diffeomorphism in the so- called algebraic case. The third appendix treats the PoincarĂ© problem on the stability of an equilibrium of a time-periodic system on \(\mathbb{R}^ 2\) in the so-called general transcendental case. The book contains few words and many formulas, and, in the reviewer’s opinion, it is rather difficult to read for a beginner although formally the book requires nothing but the standard university mathematical courses and some experience in normal forms. On the other hand, an expert will find it very useful, because the author has succeeded in gathering, in a book of less than 150 pages, many important, interesting, diverse, and often recent results on nonlinear oscillations in multidimensional systems.

34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordinary differential equations
34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
34C23 Bifurcation theory for ordinary differential equations
34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
34D10 Perturbations of ordinary differential equations
37C55 Periodic and quasi-periodic flows and diffeomorphisms
70H05 Hamilton’s equations
70K99 Nonlinear dynamics in mechanics