Asymptotic Methods for Relaxation Oscillations and Applications

TLDR: In this paper, a mathematical characterization of relaxation oscillators is presented, and a rigorous theory for weakly coupled oscillators with different limit cycles is presented. But the analysis is restricted to the case of the Van der Pol oscillator with a constant forcing term.

Abstract: 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 Poincare-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 Poincare 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.