多平衡态下簇发振荡产生机理及吸引子结构分析

以周期激励下受控Lorenz模型为例, 考察了多平衡态共存下激励频率与系统固有频率之间存在量级差距也即存在频域上的不同尺度时的耦合效应. 由于激励频率远小于系统的固有频率, 因此将整个激励项视为慢变参数, 分析随慢变参数变化下的各种分岔模式及其相应的分岔行为, 指出在一定条件下, 不同平衡点会产生Hopf分岔和fold分岔. 根据分岔条件的不同, 给出了两种典型情况下的簇发振荡, 并通过引入转换相图, 揭示了不同簇发的产生机理, 指出多平衡态和多种分岔共存不仅会导致沉寂态和激发态的多样性, 而且会使得不同沉寂态和激发态之间存在着不同的转换形式.

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

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.