On the investigation of the bifurcation and chaotic phenomena in the system with a homoclinic “figure-eight”

Small time-periodic perturbations of an asymmetric Duffing – Van der Pol equation with a homoclinic “figure-eight” of a saddle are considered. Using the Melnikov analytical method and numerical simulations, basic bifurcations associated with the presence of a non-rough homoclinic curve in this equation are studied. In the main parameter plane the bifurcation diagram for the Poincaré map is constructed. Depending on the parameters, the boundaries of attraction basins of stable fixed (periodic) points of the direct (inverse) Poincaré map are investigated. It is ascertained that the transition moment of the fractal dimension of attraction basin boundaries of attractors through the unit may be preceded by the moment of occurrence of the first homoclinic tangency of the invariant curves of the saddle fixed point.