Asymptotic methods for relaxation oscillations and applications / Johan Grasman

The book gives a survey of the theory and some of the applications of relaxation oscillations. It consists of 4 chapters. In the first one finds an introduction to the subject, numerous examples and an idea of matched asymptotic expansions. In Chapter 2, a definition of relaxation oscillation is given, existence of periodic solutions is discussed for singularly perturbed systems, asymptotic solutions of the Van der Pol oscillator and Volterra-Lotka equations are given, chemical oscillations and stochastic and chaotic oscillations are treated. In Chapter 3, forced oscillations are studied, a theory of weakly coupled oscillators is given, piecewise linear and Van der Pol (cubic) oscillators are studied, model biological oscillations under some special assumptions about coupling are discussed. In Chapter 4, the dynamics of the chaotic behavior of the solutions to some nonlinear equations is studied. There are four appendices which treat asymptotics of special functions, definitions of the symbols used in asymptotic expansions, elements of the theory of dynamical systems, and diffusion approximations for stochastic differential equations. An extensive list of references, author index and subject index are appended. The book is intended as an introduction to the field for a mixed readership. (zbMath)