Bifurcations of mixed-mode oscillations in a three-variable autonomous Van der Pol-Duffing model with a cross-shaped phase diagram

Abstract

The bifurcation structure of a three-variable Van der Pol-Duffing-type model is studied in some detail, with special attention to the mixed-mode solutions, a type of complex periodic behavior frequently encountered in oscillating chemical reactions. The mixed-mode oscillations in the model occur close to two Hopf bifurcations, which are arranged with the saddle-node bifurcations in a so-called cross-shaped phase diagram, a bifurcation diagram also typical for chemical reactions. The mixed-mode oscillations are shown to lie on isolated bifurcation curves, which are all born in a single codimension-two bifurcation known as the neutrally twisted homoclinic orbit or inclination switch. With the introduction of an additional slow time scale, the same model can exhibit more complex mixed-mode oscillations and torus bifurcations.