Asymptotic Methods for Relaxation Oscillations and Applications (Applied Mathematical Sciences, 63)

In various fields of science, notably in physics and biology, one is con­ fronted with periodic phenomena having a remarkable temporal structure: it is as if certain systems are periodically reset in an initial state. A paper of Van der Pol in the Philosophical Magazine of 1926 started up the investigation of this highly nonlinear type of oscillation for which Van der Pol coined the name "relaxation oscillation". The study of relaxation oscillations requires a mathematical analysis which differs strongly from the well-known theory of almost linear oscillations. In this monograph the method of matched asymptotic expansions is employed to approximate the periodic orbit of a relaxation oscillator. As an introduction, in chapter 2 the asymptotic analysis of Van der Pol's equation is carried out in all detail. The problem exhibits all features characteristic for a relaxation oscillation. From this case study one may learn how to handle other or more generally formulated relaxation oscillations. In the survey special attention is given to biological and chemical relaxation oscillators. In chapter 2 a general definition of a relaxation oscillation is formulated.

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.- 2.1.1 A mathematical characterization of relaxation oscillations.- 2.1.2 Application of the Poincaré-Bendixson theorem.- 2.1.3 Application of the extension theorem.- 2.1.4 Application of Tikhonov’s theorem.- 2.1.5 The analytical method of Cartwright.- 2.2 Asymptotic solution of the Van der Pol equation.- 2.2.1 The physical plane.- 2.2.2 The phase plane.- 2.2.3 The Lienard plane.- 2.2.4 Approximations of amplitude and period.- 2.3 The Volterra-Lotka equations.- 2.3.1 Modeling prey-predator systems.- 2.3.2 Oscillations with both state variables having a large amplitude.- 2.3.3 Oscillations with one state variable having a large amplitude.- 2.3.4 The period for large amplitude oscillations by inverse Laplace asymptotics.- 2.4 Chemical oscillations.- 2.4.1 The Brusselator.- 2.4.2 The Belousov-Zhabotinskii reaction and the Oregonator.- 2.5 Bifurcation of the Van der Pol equation with a constant forcing term.- 2.5.1 Modeling nerve excitation; the Bonhoeffer-Van der Pol equation.- 2.5.2 Canards.- 2.6 Stochastic and chaotic oscillations.- 2.6.1 Chaotic relaxation oscillations.- 2.6.2 Randomly perturbed oscillations.- 2.6.3 The Van der Pol oscillator with a random forcing term.- 2.6.4 Distinction between chaos and noise.- 3. Forced oscillation and mutual entrainment.- 3.1 Modeling coupled oscillations.- 3.1.1 Oscillations in the applied sciences.- 3.1.2 The system of differential equations and the method of analysis.- 3.2 A rigorous theory for weakly coupled oscillators.- 3.2.1 Validity of the discontinuous approximation.- 3.2.2 Construction of the asymptotic solution.- 3.2.3 Existence of a periodic solution.- 3.2.4 Formal extension to oscillators coupled with delay.- 3.3 Coupling of two oscillators.- 3.3.1 Piece-wise linear oscillators.- 3.3.2 Van der Pol oscillators.- 3.3.3 Entrainment with frequency ratio 1:3.- 3.3.4 Oscillators with different limit cycles.- Modeling biological oscillations.- 3.4.1 Entrainment with frequency ratio n:m.- 3.4.2 A chain of oscillators with decreasing autonomous frequency.- 3.4.3 A large population of coupled oscillators with widely different frequencies.- 3.4.4 A large population of coupled oscillators with frequencies having a Gaussian distribution.- 3.4.5 Periodic structures of coupled oscillators.- 3.4.6 Nonlinear phase diffusion equations.- 4. The Van der Pol oscillator with a sinusoidal forcing term.- 4.1 Qualitative methods of analysis.- 4.1.1 Global behavior and the Poincaré mapping.- 4.1.2 The use of symbolic dynamics.- 4.1.3 Some remarks on the annulus mapping.- 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.- 4.2.1 Subharmonic solutions.- 4.2.2 Dips slices and chaotic solutions.- 4.3 Asymptotic solution of the Van der Pol equation with a large forcing term.- 4.3.1 Subharmonic solutions.- 4.3.2 Dips and slices.- 4.3.3 Irregular solutions.- 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.