Based on the traditional scheme for a nonlinear system with multiple time scales,the enveloping slow-fast analysis method is developed in the paper,which can be employed to investigate the dynamics of nonlinear fields with multiple time scales with periodic excitation.Upon using the method,the behaviors of the kinetic model of CO oxidation on the platinum group metals have been explored in detail.Two typical bursting phenomena such as Fold/Fold/Hopf bursting and Fold/Fold bursting,are presented,the bifurcation mechanisms of which have been obtained.Furthermore,the dynamic difference between the two cases corresponding to relatively large and small perturbation frequencies,respectively,has been presented,which can be used to describe the influence of the frequencies involving in the evolution on the bursting behaviors in the system.
By introducing periodic switching signal associated with illumination to the Originator,a switched mathematical model has been established.The bifurcation sets are derived based on the characteristics of the equilibrium points.Two types of periodic oscillation,such as 2T-focus/cycle periodic switching and 2T-focus/focus periodic switching,have been observed,the mechanism of which is presented through the switching relationship.The distribution of eigenvalues related to the equilibrium points determined by two subsystems is discussed to interpret oscillation-increasing and oscillation-decreasing cascades of the periodic oscillations.Furthermore,the invariant subspaces of the equilibrium point are investigated to reveal the mechanism of dynamical phenomena in the periodic switching.
This paper investigates the generation of complex bursting patterns in Van der Pol system with two slowly changing external forcings.Complex bursting patterns,including complex periodic bursting and chaotic bursting,are presented for the cases when the two frequencies are commensurate and incommensurate.These complex bursting patterns are novel and have not been reported in previous work.Based on the fast-slow dynamics,the evolution processes of the slow forcing are presented to reveal the dynamical mechanisms underlying the appearance of these complex bursting patterns.With the change of amplitudes and frequencies,the slow forcing may visit the spiking and rest areas in different ways,which leads to the generation of different complex bursting patterns.
The mathematical model of CO oxidation with three time scales on platinum group metals is investigated, in which order gaps between the time scales related to external perturbation and the rates associated with different chemical reaction steps exist. Forced bursters, such as point-point type forced bursting and point-cycle type forced bursting, are presented. The bifurcation mechanism of forced bursting is novel, and the phenomenon where two different kinds of spiking states coexist in point-cycle type forced bursting has not been reported in previous work. A double-parameter bifurcation set of the fast subsystem is explored to reveal the transition mechanisms of different forced bursters with parameter variation.
