Nonlinear dynamical systems. (English) Zbl 0588.93001

Cook, P. A.

Prentice-Hall International Series in Systems and Control Engineering. Englewood Cliffs, New Jersey etc.: Prentice/Hall International. VII, 216 p. £24.95 (1986).

This work is intended as a textbook to be used in M. Sc. and final-year B. Sc. courses in control theory of nonlinear dynamical (deterministic) systems. While it focuses on the lectures given by the author to postgraduate students in Control Systems, it also contains useful comments on more advanced topics e.g.: singular points, limit cycles (Poincaré-Bendixson theorem), strange attractors and chaos, and so one. The book consists of: Preface, Introduction, 7 chapters, two appendices, References and Subject Index. Besides the more advanced topics above, included are: the linearisation of nonlinear functions, oscillations in feedback systems, forced systems, Lyapunov’s methods, Popop’s method, input-output methods, discrete-time systems and so one. The book also contains a number of significant worked examples which are treated from a control engineering viewpoint. At the end of the chapters 2-7 the author has also included interesting exercises (whose solutions are provided in Appendix 2). The general results, e.g. Poincaré-Bendixson theorem, are presented without proof, but information regarding the sources of the corresponding material are given in Appendix 1.

Reviewer: N.H.Pavel

MSC:

93-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to systems and control theory
93C10	Nonlinear systems in control theory
34C05	Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C15	Nonlinear oscillations and coupled oscillators for ordinary differential equations
34D20	Stability of solutions to ordinary differential equations
34D99	Stability theory for ordinary differential equations
37D45	Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37G15	Bifurcations of limit cycles and periodic orbits in dynamical systems
93C15	Control/observation systems governed by ordinary differential equations
93C55	Discrete-time control/observation systems
93C99	Model systems in control theory
93D05	Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
93D10	Popov-type stability of feedback systems
93D15	Stabilization of systems by feedback
93D25	Input-output approaches in control theory

Keywords:

singular points; limit cycles; Poincaré-Bendixson theorem; strange attractors; chaos; oscillations in feedback systems; Lyapunov’s methods; Popop’s method; input-output methods; discrete-time systems