Cluster Vibration And Bifurcation Mechanisms Of Two Kinds Of Fractional-order Differential Systems

Duffing system and Brusselator oscillator is typically nonlinear systems in the field of mechanical and chemical engineering.The actual systems are affected by various parameters.For example,fractional orders,excitation amplitude and excitation frequency change,which make these systems have more abundant nonlinear behavior.In this paper,the stability theory,bifurcation theory and central manifold theorem of differential equations are used to qualitatively analyze the complex nonlinear behavior of fractional-order Duffing system and Brusselator oscillator.The main research contents are as follows:Firstly,cluster vibration behavior and bifurcation mechanism of fractional-order Duffing system under forced excitation are studied.When the forced excitation frequency is much less than the natural frequency of the system,the whole forced excitation term can be regarded as a slow variable parameter,and the system shows a typical fast-slow phenomenon.Based on the stability and bifurcation theory of the generalized autonomous system,it is found that there is a vast difference in bifurcation behavior between the integral-order system and the fractional-order derivative system.Under the condition of given parameters,there is no Hopf bifurcation in the integer-order system,and only when the order is greater than 1,the Hopf bifurcation appears,and the corresponding proof is determined by theory.In addition,the cluster vibration behavior of the system under forced excitation is discussed,with the increase of fractional-order,the system shows abundant nonlinear behavior.They are symmetrical point-point type,symmetrical point-cycle type and symmetrical cycle-cycle type bursting.The inducing mechanism of different cluster forms is revealed.Secondly,cluster vibration behavior and bifurcation mechanism of the fractional-order Duffing system under parametric excitation are considered by using the slow-fast analysis method.By setting the amplitude of the parametric excitation,the bifurcation delay is terminated in different regions of the bifurcation diagram of the fast subsystem,and different forms of vibration appear.When the system is only affected by the equilibrium point,because of the occurrence of fork type bifurcation,it shows as point-point type.When the parameter excitation increases,two Hopf bifurcations appear,and the corresponding two limit cycles affect the trajectory of the system.Due to the influence of the bifurcations and Hopf bifurcations,it appears as a point-cycle type.As the excitation continues to grow,two limit cycles collide to produce a large limit cycle.Meanwhile,the system is affected by bifurcation.Hopf bifurcation and limit cycle bifurcation,which are represented by point-cycle-cycle type bursting.At the same time,the influence of parameters on cluster vibration is discussed by adjusting system parameters and the corresponding mechanism explanation is given.Finally,cluster oscillation behavior and bifurcation of fractional-order Brusselator oscillator under periodic excitation are discussed.The stability theory and bifurcation theory of fractional-order system is used to study the cluster behavior in detail.It is noted that with the decrease of fractional-order,the distance between two Hopf bifurcation points becomes smaller,which leads to the shortening of the time when the system is in the excited state.The consistency between the numerical simulation and theoretical analysis is further illustrated by the transition phase diagram of fractional-order system and the superposition diagram of the bifurcation diagram.Under the condition of certain parameters,the system has the phenomenon of double Hopf periodic cluster oscillation.The mechanism is the case that the change of the amplitude of the external periodic disturbance makes the type and number of attractors involved in the fast subsystem of the system change.The induced mechanism is further revealed by the superposition of the transformation phase diagram and the bifurcation diagram.