Mechanism of bursting oscillations with multiple equilibrium states and the analysis of the structures of the attractors

Abstract

The main purpose of this article is to explore the bursting behaviors as well as the mechanism when multiple equilibrium states evolve into the bursting attractors. Taking the controlled Lorenz model with periodic excitation for example, the coupling effect of different scales in frequency domain corresponding to the case that an order gap exists between the exciting frequency and the natural frequency of the system with multiple equilibrium states is investigated. Unlike the autonomous slow-fast coupling system, neither obvious slow nor fast subsystems can be observed in a periodically excited system. Since the exciting frequency is far less than the natural frequency of the system, the whole exciting term can be considered as a slow-varying parameter, leading to the generalized autonomous system. With the variation of the slowly-varying parameter, the bifurcation forms as well as the behaviors for different equilibrium states in the generalized autonomous system are explored. It is pointed out that for certain conditions, Hopf bifurcation and fold bifurcations related to different equilibrium points can be observed. According to the conditions related to different bifurcations, the bursting oscillations in two typical cases are presented. In order to explore the mechanism of bursting oscillation, transformed phase portraits are introduced in which the whole exciting term is treated as a generalized state variable so that the relationship between the original state variables and the slow-varying parameter can be clearly described. By employing the transformed phase portraits, the bifurcation mechanisms of different bursting attractors are presented. For the conditions where only fold bifurcation exists between two equilibrium states in the generalized autonomous system, two un-symmetric bursting attractors can be observed. With the variation of parameters, when the repetitive spiking oscillations pass across the attracting basin of another equilibrium states, the two bursting attractors interact with each other to form an enlarged symmetric bursting attractor. For the conditions where both the Hopf and fold bifurcations evolve into the bursting attractors, multiple quiescent states as well as repetitive spiking states exist in the bursting oscillations, which may lead to complicated behaviors. It is found that the coexistence of multiple equilibrium states as well as the related bifurcation forms not only leads to multiple forms of quiescent states and the spiking states, but also results in different switching forms between different quiescent states and the spiking states.