Singular Perturbations and a Mapping on an Interval for the Forced van der Pol Relaxation Oscillator

摘要

The Van der Pol relaxation oscillator with a large sinusoidal forcing term is discussed. By using singular perturbation techniques, asymptotic solutions of such a system are constructed. These asymptotic approximations are locally valid and take the form of a two time scale expansion in one region and a boundary layer type of solution in the 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, which is an integrated mapping on a compact interval and solutions of the system itself. This equivalence makes it possible to analyze subharmonics and chaotic type solutions to the full extent. Domains in the parameter space where regular subharmonics exist are found. These domains overlap so that for some parameter values, different subharmonics coexist. For the same values, chaotic type solutions are found and are described by using concepts of symbolic dynamics.