Research On Extreme Multi-stability Of Four-dimensional Chaotic System With Discrete Bifurcation Diagrams

In recent years,the multi-stability and extreme multi-stability of chaotic systems have become a new research focus in the field of chaotic systems.The multi-stability of chaotic systems can be applied to image processing or to generate different random signal sources for many information engineering fields,such as information encryption,secure communication,etc.Therefore,studying the multiple stability of chaotic systems has important practical significance and application value.As the fourth basic circuit element,memristors are used to design chaotic systems with complex dynamics due to their nonlinearity,low power consumption and nanometer size.On the basis of reading a large number of literatures and materials,this paper proposes a four-dimensional memristive chaotic system with discrete bifurcation diagrams and a four-dimensional chaotic system with infinitely many symmetrical homogeneous attractors,both of which possess multi-stability and extreme multi-stability.The specific research content and results achieved are as follows:Firstly,a flux-controlled memristor is introduced into a three-dimensional chaotic system,and a new four-dimensional memristive chaotic system is proposed.The basic dynamic characteristics of the system are analyzed by numerical simulation.The system can generate single,double and four-scroll chaotic attractors.Then using bifurcation diagrams,Lyapunov exponential spectrum and other methods to analyze the influence of the initial value of the system on the system,it is found that the system has homogeneous multi-stability,heterogeneous multi-stability and extreme multi-stability.In addition,the system also has a wide range of initial value changes,discrete bifurcation diagrams and infinitely many symmetrical coexisting attractors.Finally,the PSpice circuit simulation software is used to build a circuit to verify the system.The circuit simulation results are in good agreement with the Matlab numerical simulation results.Then,on the basis of a classic three-dimensional chaotic system,a new four-dimensional chaotic system with infinitely many symmetric homogeneous attractors is proposed.This system does not use memristors,but it also has a line equilibrium that can generate hidden attractors.The system is analyzed using phase orbit diagrams,bifurcation diagrams and Lyapunov exponential spectrum.The results show that the system has constant Lyapunov exponent spectrum except for the zero point,centrally symmetric discrete bifurcation diagrams,and infinitely many symmetrical homogeneous attractors.Finally,a simulation circuit was built according to the system equations to verify the system by experiments.The PSpice circuit simulation results verified the correctness of the previous Matlab numerical simulations.