Modeling critical phenomena in a generalized van der Pol model

Abstract

The van der Pol model has a long history of being used in both the physical and biological sciences. This paper considers modeling critical phenomena in one of three-time scales fast-slow autonomous dynamical systems. We investigate a model that is a generalization of the classical van der Pol model. In the framework of this paper, we construct an invariant manifold with variable stability for the generalized van der Pol model. Expansions of the manifold and corresponding gluing function were constructed using the flow curvature method. We show the existence of a sufficiently smooth invariant surface (a black swan surface) for the model. Computer algebra methods are used for the quantitative analysis of the model.