BIFURCATIONS OF MIXED-MODE OSCILLATIONS IN A 3-VARIABLE AUTONOMOUS VANDERPOL-DUFFING MODEL WITH A CROSS-SHAPED PHASE-DIAGRAM

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 encount ered in oscillating chemical reactions. The mixed-mode oscillations in the model occur close to two Hopf bifurcations, which are arranged wi th the saddle-node bifurcations in a so-called cross-shaped phase diag ram, a bifurcation diagram also typical for chemical reactions. The mi xed-mode oscillations are shown to lie on isolated bifurcation curves, which are all born in a single codimension-two bifurcation known as t he neutrally twisted homoclinic orbit or inclination switch. With the introduction of an additional slow time scale, the same model can exhi bit more complex mixed-mode oscillations and torus bifurcations.