Bifurcations for heteroclinic orbits of a periodic motion and a saddle-focus and dynamical chaos

Bifurcations and the structure of limit sets are studied for a three-dimensional van der Pol-Duffing system with a cubic non-linearity. On the basis of both computer simulations and theoretical results, a model map is proposed which allows one to follow the evolution in the phase space from a simple (Morse-Smale) structure to chaos. It is established that the appearance of complex, multistructural sets of double-scroll type is stipulated by the presence of a heteroclinic orbit of intersection of the unstable manifold of a saddle periodic orbit and stable manifold of an equilibrium state of saddle-focus type.