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.
The dynamics of a typical Belousov-Zhabotinsky(BZ)reaction with multiple time scales is investigated in this paper.Different forms of periodic bursting phenomena,and specially,three types of chaotic bursters with different structures can be obtained,which are in common with the behaviors observed in experiments.The bifurcations connecting the quiescent state and the repetitive spikes are presented to account for the occurrence of the NKoscillations as well as the different forms of chaotic bursters.The mechanism of the period-adding bifurcation sequences is explored to reveal why the length of the periods in the sequences does not change continuously with the continuous variation of the parameters.
BI QinSheng Faculty of Science,Jiangsu University,Zhenjiang 212013,China
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 behaviors of a system that alternates between the R¨ossler oscillator and Chua's circuit is investigated to explore the influence of the switches on the dynamical evolution.Switches related to the state variables are introduced,upon which a typical switching dynamical model is established.Bifurcation sets of the subsystems are derived via analysis of the related equilibrium points,which divide the parameters into several regions corresponding to different types of attractors.The dynamics behave typically in period orbits with the variation of the parameters.The focus/cycle periodic switching phenomenon is explored in detail to present the mechanism of the movement.The period-doubling bifurcation to chaos can be observed via the doubling increase of the turning points related to the switches.Furthermore,period-decreasing sequences have been obtained,which can be explained by the variation of the eigenvalues associated with the equilibrium points of the subsystems.
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.
The dynamics of a non-smooth electric circuit with an order gap between its parameters is investigated in this paper.Different types of symmetric bursting phenomena can be observed in numerical simulations.Their dynamical behaviours are discussed by means of slow-fast analysis.Furthermore,the generalized Jacobian matrix at the non-smooth boundaries is introduced to explore the bifurcation mechanism for the bursting solutions,which can also be used to account for the evolution of the complicated structures of the phase portraits.With the variation of the parameter,the periodic symmetric bursting can evolve into chaotic symmetric bursting via period-doubling bifurcation.
