分数阶Brusselator振子的簇发振动与分岔

Cluster vibration and bifurcation of a fractional-order Brusselator oscillator

由于催化剂的存在,Brusselator振子是典型的多尺度耦合系统,即常常存在激发态和沉寂态耦合的簇发振动行为。考虑分数阶Brusselator系统的催化过程受到外部周期扰动下的情形,这使系统的非线性行为更加复杂。根据分数阶系统稳定性理论进行了双参数分岔分析,讨论了Hopf分岔的充分条件。发现系统存在一条奇线,利用中心流形定理和数值模拟验证了该奇线的稳定性。探讨了分数阶阶次对簇发振动的影响,通过分数阶阶次与慢变参数的双参数分岔图,发现分数阶阶次与激发态时间长短密切相关,即降低分数阶阶次,可以缩短激发态时间,从而增加沉寂态的时间。研究还发现扰动幅值的变化直接影响快子系统的吸引子类型,当激励幅值较大时,快子系统涉及到两种吸引子,沉寂态和激发态并存;当激励幅值较小时,快子系统涉及一种吸引子,沉寂态基本消失。

A Brusselator oscillator is a typical multi-scale coupling system because of catalyst,which will lead to cluster vibration behavior,characterized by the spiking state coupled with the quiescence state.In this paper,we consider the fractional-order Brusselator system under external periodic disturbance,and the nonlinear behaviors of the system are more complex.Based on the stability theory of a fractional order system,the two-parameter bifurcation analysis was carried out,and the sufficient conditions of Hopf bifurcation were discussed.It was found that there is a singular line in the system,and its stability was verified by using the center manifold theorem and numerical simulation.The influence of different fractional orders on cluster vibration was discussed.Through the two-parameter bifurcation diagram with respect to fractional order and slowly varying parameters,it was found that the fractional order is closely related to the time of the spiking state.That is to say,reducing the fractional order of the system can shorten the time of the spiking state and increase the time of the quiescence state.It was also found that the variation of disturbance amplitude directly affects the type of the attractor of the fast subsystem.When the excitation amplitude is large,two kinds of attractors are involved in the fast subsystem,the quiescence state and the spiking state coexist.When the excitation amplitude is small,the fast subsystem involves one kind of the attractor,then the quiescence state disappears.