Local Regularity in Chaotic Bursting Oscillations Induced by Stable Attractors in Fast Subsystem

Chaos is thought to be random, but a few constants, such as Lyapunov exponent and fractal dimension, can be obtained by statistic measures based on a large amount of data. This paper presents the local regularity in chaotic bursting oscillations caused by the coupling of two scales in frequency domain. When slow-varying external excitation is introduced in a well-known Nosé–Hoover oscillator, called also Sprott-A system, different types of bursting attractors, such as periodic and chaotic bursting oscillations, can be observed. By regarding the external excitation as a slow-varying parameter, the dynamical evolution of the generalized autonomous fast subsystem is obtained. Two typical cases for the physical parameters are considered, in which codimension-1 fold and codimension-2 fold–Hopf bifurcations take place, respectively. Several types of coexistent attractors in the fast subsystem, such as the coexistence of equilibrium points, the coexistence of equilibrium point and sequence of period-doubling bifurcation routine to chaos, may appear, which may lead to different bursting oscillations, the mechanism of which is presented by overlapping the transformed phase portrait and the behaviors of the fast subsystem. The trajectory of periodic bursting oscillations visits the equilibrium points in turn via fold bifurcations, while fold–Hopf bifurcation may result in quasi-periodic bursting oscillations. Two coexisted asymmetric bursting attractors may expand in the phase space with the increase of exciting amplitude. The interaction forms an enlarged symmetric bursting attractor. The influence of the periodic windows on the full vector field may appear, which causes the spiking stage to be divided into segments with different behaviors. Specially spiking state can oscillate according to different regular cycles as well as chaos in turn, which suggests that regular attractors are embedded in chaos. More periodic windows exist in the fast subsystem, more fine structures can be observed in the spiking stage. Accordingly, the coexisted attractors, together with the related bifurcations, in the fast subsystem, may be one of the main reasons for the complexity of the world with the coupling of multiple scales.