Fine Structures in Chaotic Bursting Attractors Induced by Slow–Fast Dynamics with Different Kinds of Nonlinearity

The coupling of multiple scales in nonlinear dynamics can often lead to bursting oscillations, in which chaotic bursting oscillation may be one of the most complicated behaviors. The main purpose of the paper focuses on the scale effect of the bursting oscillations. By introducing the combination of parametric and external excitations with a small exciting frequency in the typical Chua’s oscillator, evolution from quasi-periodic bursting attractor to chaos can be observed with the variation of the exciting amplitude. Bifurcation analysis of the generalized autonomous fast subsystem reveals that, for the cubic nonlinearity, a chaotic attractor via heteroclinic bifurcation coexists with a large-amplitude cycle, causing the trajectory of chaotic bursting oscillations in spiking state to change between chaos and quasi-periodic vibration, which is demonstrated by the projections of the Poincaré map. However, for the fifth-order nonlinearity, sequence of period-doubling bifurcations to chaos can be observed in the fast subsystem, resulting in a set of periodic windows in the chaotic region. The chaotic spiking trajectory alternates between chaos and a set of tori corresponding to the periodic windows. Since all the lengths of the periodic windows along the slow-varying parameter axis are relatively short, the influence of the windows on the slow–fast dynamics becomes clearer with the decrease of the ratio between the scales, because of the increase of movement inertia along the windows, which yields finer structures in the chaotic bursting attractor. It suggests that in the chaotic dynamics induced by a set of different scales, the movement may behave in regular oscillations and chaos in different regions of the phase space, though the whole trajectory is chaotic, implying order exists in disorder, in which regular attractors seem to be embedded in chaotic attractors.