Abstract
The integer-order and fractional-order Brusselators with two different time scales are studied. The double-Hopf bursting oscillation is observed in Brusselator with periodic perturbation under some parameter conditions. Based on slow-fast analysis and bifurcation theory, the generation mechanism of periodic bursting oscillation is presented in detail. Further investigation finds that the perturbation amplitude plays an important role on bursting oscillation. With the decrease of perturbation amplitude, the attractor types will be changed, so that the bi-stability evolves into single stability. It causes that four times transitions between spiking and quiescent states may decrease to twice one, and the spiking state disappears finally. Furthermore, the influence of the fractional order on bursting behavior is investigated, and the bifurcation diagram with respect to fractional order and slow variable is given. With the decrease of fractional order, the two Hopf bifurcation points may approach each other gradually, overlap and disappear finally, which results into the transition from periodic bursting oscillation with different frequency ingredients into generally periodic oscillation with single frequency.
Highlights
The Brusselator model for autocatalytic oscillating chemical reactions was introduced by Prigogine and Lefever in 1968 [1]
When the external periodic disturbance frequency is much smaller than the system natural frequency, the system has two different time scales and shows an obvious slow-fast effect
The Hopf bifurcation of fast subsystem (FS) leads to transitions between spiking and quiescent states
Summary
The Brusselator model for autocatalytic oscillating chemical reactions was introduced by Prigogine and Lefever in 1968 [1]. In 1985, the slow-fast analysis method was proposed by Rinzel [10], which could effectively reveal the bifurcation mechanism of bursting oscillation. Many bursting oscillations in the multi-time scale systems were studied in different fields. BURSTING OSCILLATION AND BIFURCATION MECHANISM IN FRACTIONAL-ORDER BRUSSELATOR WITH TWO DIFFERENT TIME SCALES. We will focus on the fractional-order Brusselator with two time scales, and the novel bursting phenomenon associated with its bifurcation mechanism are investigated in detail. One can assume that external perturbation directly affects the concentration of inhibitor in the reaction process In this case, the dynamic model is considered as follows:. If the perturbation frequency is much smaller than the natural frequency of the original system, there are two time scales in the system Eq (2) with a larger gap in terms of quantity, which will lead to more complicated nonlinear dynamical behaviors
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