On occurrence of sudden increase of spiking amplitude via fold limit cycle bifurcation in a modified Van der Pol–Duffing system with slow-varying periodic excitation

The main purpose of the paper is to reveal mechanism certain special phenomena in bursting oscillations such as sudden increase spiking amplitude. When multiple equilibrium points coexist a dynamical system, several types stable attractors via different bifurcations from these may be observed with variation parameters, which interact each other form bifurcations. Here we take modified van der Pol–Duffing system an example, periodic parametric excitation introduced. exciting frequency far less than natural frequency, appear. By regarding term slow-varying parameter, number branches fast generalized autonomous subsystem varies one five parameter. undergo bifurcations, Hopf and pitchfork limit cycles, including cycles near homo-clinic orbit, fold cycle With amplitude, involve full leading oscillations. Fold cause change since at bifurcation points, trajectory oscillate according obviously amplitudes. At point, two possible jumping ways result coexisted asymmetric attractors, expand phase space enlarged symmetric attractor doubled period. inertia movement along pass across related delay effect Not only large but also value movement, both cases, rate parameter increase. Therefore, relative small taken order show influence all their on dynamics system.