Complex bursting patterns and their bifurcation mechanisms in the Φ6-van der Pol system with external and parametric excitations

Complex bursting patterns and the bifurcation generation mechanism are investigated in the parametrically and externally driven Phi(6)-van der Pol system. Five different bursting types, i.e. symmetric 'subHopf/homoclinic' bursting, symmetric 'subHopf/fold-cycle' bursting via 'subHopf/subHopf hysteresis loop, symmetric 'subHopf/fold-cycle' bursting, 'subHopf/subHopf' bursting and 'fold/fold' bursting, are explored. First, the mechanism of symmetric 'subHopf/fold-cycle' bursting via 'subHopf/subHopf' hysteresis loop is revealed. Then, based on the two-parameter bifurcation analysis, dynamical evolutions of the symmetric bursting are presented. Different clusters appear when the parameter changes, leading to different patterns of bursting oscillations. In addition, we find that the jumping phenomenon depends not only on the fold bifurcation, but also on the subcritical Hopf bifurcation. Finally, numerical simulations are presented to demonstrate the correctness of the research.