The Second-Order Melnikov Function and Chaotic Tangles on Toral van der Pol Equation

Abstract

The van der Pol equation on the torus is considered. This equation contains a heteroclinic cycle consisting of four symmetric heteroclinic orbits. By high order Melnikov method, the periodic forced van der Pol equation is investigated and chaotic dynamics is obtained. An explicit formula of the second-order Melnikov function is derived for the splitting heteroclinic connection. It is used to detect chaotic dynamics when the first-order(classical) Melnikov function is degenerate. By the second-order Melnikov function, we deduce chaotic heteroclinic tangles with rigorously theoretical analysis as well as numerical simulations.