Research On Chaotic Dynamics Of Jerk Systems

Chaos is random-like behavior occurring without adding any random factors in a deterministic nonlinear system.The greatest characteristic of chaotic motion is its extreme sensitivity to the initial values and parameters of the system.In recent years,with the rapid development and widespread application of computer technology,the chaotic phenomena in nonlinear systems have become a hot topic in chaos science all over the world.The first problem of research on chaos is finding nonlinear systems whose equations are as simple as possible and easy to implement in real world.More and more researchers paid attentions to the Jerk system in the nonlinear field,because of its simple equation form and easy circuit implementation.As an extension of Jerk system,Hyperjerk system also has simple algebraic structure and more complex dynamic characteristics than Jerk system.Therefore,the research of Jerk system and Hyperjerk system are of great significance.In this paper,the chaotic dynamics of Jerk system and Hyperjerk system are studied by means of theoretical analysis,numerical simulation,and circuit design.The specific research results are as follows:(1)Research on chaotic dynamics of the new three-dimensional autonomous Jerk system: By designing a three-dimensional autonomous Jerk chaotic system and analysing its dynamic behaviors.We found that the system has the following four significant characteristics: 1)The system contains three cubic terms and only one equilibrium point at the origin;2)This is a typical example of the existence of asymmetric bistability(e.g.coexistence of a point attractor and chaotic attractor or coexistence of a point attractor and period-5 limit cycle)in a symmetric chaotic system,that is,the system equation is symmetric about the coordinate origin,but there are asymmetric coexistence attractors in the system;3)When we selected the appropriate system parameters and initial conditions,the system exhibited symmetrical bistability and tristability(e.g.coexistence of a point attractor and a pair of symmetric chaotic or periodic attractors),period-doubling bifurcation,reverse period-doubling bifurcation and antimonotonicity;4)There are coexisting bubbling and attractors merging in the system,which is rarely reported before.In addition,we also designed the equivalent circuit and use the hardware circuit to realize.The numerical simulation results,the simulation results of the Multisim circuit and the test results of the hardware circuit are identical.(2)Research on chaotic dynamics of improved Hyperjerk system without equilibrium: An improved Hyperjerk chaotic system with no-equilibrium is obtained by improving the Hyperjerk chaotic system with a stable equilibrium.Nonlinear analysis tools such as bifurcation diagram,Lyapunov exponential spectrum,phase diagram,Poincaré sections,0-1 test diagram and time series diagram are used to analysed the complex dynamic behaviors of the system in detail.In particular,the system also has the grid offset boosting behaviors,so far,the grid offset boosting behavior has not been reported in other literature.Finally,the circuit is designed and the actual circuit is built with off-the-shelf electronic components.The correctness of the theoretical analysis results is verified by the digital oscilloscope.