Non-standard reduction of noisy Duffing—van der Pol equation

Abstract

The purpose of this work is to improve the understanding of the solution of the noisy Duffing-van der Pol equation. We achieve this by developing rigorous methods to replace, in some limiting regime, the original complicated system by a 'simpler, constructive, and rational' approximation - a low-dimensional model of the dynamical system. To this end, we study the equations as a random perturbation of a two-dimensional Hamiltonian system. We achieve the model-reduction through 'stochastic averaging' and the reduced Markov process takes its values on a graph with certain glueing conditions at the vertex of the graph. Examination of the reduced Markov process on the graph yields many important results, namely, mean exit times, probability density functions, and stochastic bifurcations.