Singular perturbations and a mapping on an interval for the forced Van der pol relaxation oscillator

Abstract This paper deals with the Van der Pol relaxation oscillator with a large sinusoidal forcing term. By using singular perturbation techniques asymptotic solutions of such a system are constructed. These asymptotic approximations are locally valid and may take the form of a two time scale expansion in one region and a boundary layer type of solution in a next region. Integration constants are determined by averaging and matching conditions. From these local solutions a difference equation is constructed. There is an equivalence between solutions of the difference equation being an iterated mapping on a compact interval and solutions of the system itself. This equivalence makes it possible to analyze subharmonics and chaotic type of solutions to the full extent. As a result of this we find domains in the parameter space, where regular subharmonics exist. These domains overlap so that for some parameter values different subharmonics coexist. For parameter values in this range chaotic type of solutions are found as well. They are described by using concepts of symbolic dynamics.

[1]  S. Smale.  Diffeomorphisms with Many Periodic Points , 1965 .

[2]  J. E. Littlewood,et al.  On non-linear differential equations of the second order: III. The equation $$\ddot y - k(1 - y^2 )\dot y + y = b \mu k cos (\mu t + \alpha )$$ for largek, and its generalizationsfor largek, and its generalizations , 1957 .

[3]  J. Eckmann,et al.  Iterated maps on the interval as dynamical systems , 1980 .

[4]  J. Grasman.  Relaxation oscillations of a van der pol equation with large critical forcing term : (preprint) , 1978 .

[5]  J. Yorke,et al.  Period Three Implies Chaos , 1975 .

[6]  S. Smale.  Differentiable dynamical systems , 1967 .

[7]  M. L. Cartwright.  On non-linear differential equations of the second order , 1949 .

[8]  M. Levi.  Qualitative Analysis of the Periodically Forced Relaxation Oscillations , 1981 .

[9]  J. Gollub,et al.  Periodicity and Chaos in Coupled Nonlinear Oscillators , 1978, Science.

[10]  P. Stefan.  A theorem of Šarkovskii on the existence of periodic orbits of continuous endomorphisms of the real line , 1977 .

[11]  J. Grasman,et al.  Relaxation Oscillations Governed by a Van der Pol Equation with Periodic Forcing Term , 1976 .

[12]  Michael Shub,et al.  Stabilité globale des systèmes dynamiques , 1978 .

[13]  John Guckenheimer,et al.  Symbolic dynamics and relaxation oscillations , 1980 .

[14]  Johan Grasman,et al.  Asymptotic Methods for Relaxation Oscillations , 1978 .

[15]  D. Saari,et al.  Stable and Random Motions in Dynamical Systems , 1975 .

[16]  N. Levinson,et al.  A Second Order Differential Equation with Singular Solutions , 1949 .

[17]  J. Flaherty,et al.  Frequency Entrainment of a Forced van der pol Oscillator. , 1977 .

[18]  E. Lorenz.  Deterministic nonperiodic flow , 1963 .

[19]  Robert M. May,et al.  Simple mathematical models with very complicated dynamics , 1976, Nature.

[20]  H. Weitzner,et al.  Perturbation Methods in Applied Mathematics , 1969 .