Dynamics and bifurcations.

*(English)*Zbl 0745.58002
Texts in Applied Mathematics. 3. New York etc.: Springer-Verlag. xiv, 568 p., 314 ill. (1991).

This is an introductory text-book to the qualitative theory of ordinary differential and difference equations. “Unusual in both content and style” (see Greetings by the authors), it presents ideas and examples rather than formal exposition of topics. As to the last ones, the main goal of the book is to study the basic notions and examples of bifurcation theory. Thus, systems and maps depending on parameters are generally under consideration. For further exploration, see “Methods of Bifurcation Theory” (1982; Zbl 0487.47039) by S. N. Chow and the first author.

Trying to arrange the presentation to be accessible for the beginners the authors mostly concentrate themselves on low dimensions (one and two, even two-dimensional periodic in time case is considered to be “high- dimensional”). Nevertheless, starting by scalar autonomous differential equations and, respectively, by scalar difference equations, the authors succeeded in exposition of the principle ideas and notions of bifurcation theory as well as of the facts in ordinary differential equations which this theory is based upon.

The following list of covered topics characterizes the contents of the book: Hopf bifurcation, period-doubling, pitchfork, saddle-node bifurcations of equilibria and fixed points; chaos, strange attractors; homoclinic and heteroclinic orbits and points; structural stability; unfoldings.

Also Hamiltonian and gradient systems, Mathieu, Duffing, Van-der-Pol, Lorenz equations; area-preserving, Cremona, Hénon, logistic maps.

A great number of exercises and numerous illustrations enrich very much the presentation of the topics.

Trying to arrange the presentation to be accessible for the beginners the authors mostly concentrate themselves on low dimensions (one and two, even two-dimensional periodic in time case is considered to be “high- dimensional”). Nevertheless, starting by scalar autonomous differential equations and, respectively, by scalar difference equations, the authors succeeded in exposition of the principle ideas and notions of bifurcation theory as well as of the facts in ordinary differential equations which this theory is based upon.

The following list of covered topics characterizes the contents of the book: Hopf bifurcation, period-doubling, pitchfork, saddle-node bifurcations of equilibria and fixed points; chaos, strange attractors; homoclinic and heteroclinic orbits and points; structural stability; unfoldings.

Also Hamiltonian and gradient systems, Mathieu, Duffing, Van-der-Pol, Lorenz equations; area-preserving, Cremona, Hénon, logistic maps.

A great number of exercises and numerous illustrations enrich very much the presentation of the topics.

Reviewer: Yu.N.Bibikov (St.Petersburg)