Dynamic systems on measure chains. (English) Zbl 0869.34039

Lakshmikantham, V.; Sivasundaram, S.; Kaymakcalan, B.

Mathematics and its Applications (Dordrecht). 370. Dordrecht: Kluwer Academic Publishers. x, 285 p. (1996).

This book is designed to provide a unified treatment of basic dynamics of various types of dynamical systems, including, but not limited to, maps and flows. In this treatment, two basic notions are time scales and measure chains. A time scale is any closed subset of \(\mathbb{R}\); the dynamical systems considered are actions of the form \(T\times X\rightarrow X\) where \(T\) is a time scale, and \(X\) is the phase space of the dynamics. A measure chain is a particular way of measuring signed distances between points of \(T\); it is used to make it possible to describe a generalized notion of ‘differentiability’ with respect to \(t\in T\) of the dynamical system. The book consists of 4 chapters, and is largely devoted to collecting and correcting results on dynamics of measure chains that have appeared in the literature. The first chapter contains the basic definitions, and develops the analogues of undergraduate advanced calculus in the context of measure chains. Chapter 2 continues this development, extending the basic results of the theory of ordinary differential equations to measure chains. Chapter 3 describes Lyapunov stability theory in this context. The final chapter, which comprises almost half the book, is concerned with applying the developmental material in the previous chapters to a number of problems. Among the topics discussed are quasilinearization, monotone flows, invariant manifolds, problems from control, boundary value problems, and problems related to convexity.

Reviewer: M.Hurley (Cleveland)

MSC:

37-XX	Dynamical systems and ergodic theory
93-02	Research exposition (monographs, survey articles) pertaining to systems and control theory
34-02	Research exposition (monographs, survey articles) pertaining to ordinary differential equations
39-02	Research exposition (monographs, survey articles) pertaining to difference and functional equations
93D05	Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
34D20	Stability of solutions to ordinary differential equations
34H05	Control problems involving ordinary differential equations

Keywords:

dynamical systems; time scale; measure chains; Lyapunov stability theory