Asymptotic Methods for Relaxation Oscillations and Applications

1. Introduction.- 1.1 The Van der Pol oscillator.- 1.2 Mechanical prototypes of relaxation oscillators.- 1.3 Relaxation oscillations in physics and biology.- 1.4 Discontinuous approximations.- 1.5 Matched asymptotic expansions.- 1.6 Forced oscillations.- 1.7 Mutual entrainment.- 2 Free oscillation.- 2.1 Autonomous relaxation oscillation: definition and existence.- 2.2 Asymptotic solution of the Van der Pol equation.- 2.3 The Volterra-Lotka equations.- 2.4 Chemical oscillations.- 2.5 Bifurcation of the Van der Pol equation with a constant forcing term.- 2.6 Stochastic and chaotic oscillations.- 3. Forced oscillation and mutual entrainment.- 3.1 Modeling coupled oscillations.- 3.2 A rigorous theory for weakly coupled oscillators.- 3.3 Coupling of two oscillators.- 4. The Van der Pol oscillator with a sinusoidal forcing term.- 4.1 Qualitative methods of analysis.- 4.2 Asymptotic solution of the Van der Pol equation with a moderate forcing term.- 4.2 Asymptotic solution of the Van der Pol equation with a large forcing term.- 4.3 Asymptotic solution of the Van der Pol equation with a large forcing term.- Appendices.- A: Asymptotics of some special functions.- B: Asymptotic ordering and expansions.- C: Concepts of the theory of dynamical systems.- D: Stochastic differential equations and diffusion approximations.- Literature.- Author Index.