## Asymptotic methods for relaxation oscillations and applications

Johan Grasman

The book deals with the symptotic analysis of relaxation oscillations, which are nonlinear oscillations characterized by rapid change of a variable within a short time interval of the cycle. The type of asymptotic approximation of the solution is known as the method of matched asymptotic expansions. In case of coupled oscillations it gives conditions for entrainment. For spatially distributed oscillators phase wave solutions can be constructed. The asymptotic theory also covers the chaotic dynamics of free and forced oscillations. The influence of stochastic perturbations upon the period of the oscillation is also covered. It is the first book on this subject which also provides a survey of the literature, reflecting historical developments in the field. Furthermore, relaxation oscillations are analyzed using the tools drawn from modern dynamical system theory. This book is intended for graduate students and researchers interested in the modelling of periodic phenomena in physics and biology and will provide a second knowledge of the application of the theory of nonlinear oscillations to a particular class of problems.

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[目次]

• 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.

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書名 Asymptotic methods for relaxation oscillations and applications Grasman, Johan Grasman Johan (Agricultural University Wageningen The Netherlands) Relaxation oscillations and applications Applied mathematical sciences Springer-Verlag c1987 Softcover reprint of the original 1st ed. 1987 xiii, 221 p. 24 cm 3540965130 0387965130 BA00608477 ※クリックでCiNii Booksを表示 英語 アメリカ合衆国
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