Dynamical Analysis And Bifurcation Research Of Two Types Of Chaotic Systems

Chaotic systems and circuits are the focus of research in the field of nonlinear dynamics.In addition to conventional chaotic systems,local active memristors can simulate the function of neurons with complex and rich dynamical properties.A large number of bifurcations have been found in the fields of biology,physics,mathematics,and economics.A series of theories and methods have been established to analyze the bifurcation problem qualitatively.Bifurcation causes a change in the motion state of chaotic systems,which is not only closely related to the generation of chaos,but also a mechanism linking periodic motion,equilibrium state,and chaotic motion.In this paper,combining with the current chaos theory,the focus is on the dynamics analysis and bifurcation mechanism of simple jerk system,memristor jerk system and hyperchaotic system and other related issues.Based on numerical analysis methods,bifurcation theory and circuit theory,the characteristics of chaotic systems are analyzed and the bifurcation mechanism of chaotic systems is revealed,mainly as follows.(1)The zero-Hopf bifurcation at a singular point of a parameter jerk system is characterized.By employing the second order averaging theory,we demonstrate that up to three periodic orbits generated as disturbance parameters tend to zero.Again,both the bifurcation mechanism and bursting dynamics of the 3D jerk system with external periodic excitation are systematically explored.While an order difference exists between the frequency of external excitation and the average frequency of the system,the system exhibits bursting oscillations.The mechanisms of different bursting oscillations are investigated employing the equilibrium point curve and the transformed phase portraits.As the amplitudes of the excitation change,the system displays “delayed symmetric pitchfork/point bursting,delayed symmetric pitchfork/sup Hopf bursting,and delayed pitchfork/sup Hopf/homoclinic connection bursting”.Finally,an analog circuit is designed to verify the complex bursting phenomena of the system.(2)Based on the Chua expansion theorem,a locally active memristor model is proposed and introduced into the jerk system.The non-volatile and locally activity of the memristor is analyzed using power-off plot(POP)and DC V-I plot.The system generates up to four Hopf bifurcation points under different parameters.The unstable saddle focal points of index 2(USFP-2)causes the system to generate four-scroll chaotic attractors,and the mechanism of scroll generation is analyzed in detail.The system multi-stability and symmetric dynamics phenomena are clearly described using the local basins of attraction.The circuit simulation results verify the dynamic behavior of the system.(3)Chaos generation depends on nonlinear elements and nonlinear terms in systems of ordinary differential equations.A simple four-dimensional hyperchaotic system with only two nonlinear terms is proposed.The complex dynamical behaviors of the system are demonstrated with phase portraits,Lyapunov exponents spectrum,bifurcation diagram,Poincaré map and Kaplan-Yorke dimension.Again,the numerical simulation results indicate that the polarity of chaotic signals is transformed flexibly by the offset boosting.The system generates periodic bursting oscillations similar to the behavior of neurons,and the bifurcation mechanism of periodic bursting oscillations is revealed in detail by constructing the fold and Hopf bifurcation set of fast-scale systems.The proposed system exhibits symmetric periodic fold/Hopf bursting oscillations with the change of parameters.Finally,hardware experiments verify the theoretical analysis and numerical simulation of the periodic bursting.Its potential value in engineering applications is further demonstrated.