Research And Design Of Fractal Multi-scroll Memristive Chaotic System

Nonlinear science is a basic discipline studying the commonness of nonlinear phenomena,and it includes three branches of Chaos,fractal and soliton.Fractal has some basic characteristics such as irregularity,fine structure,and certain self-similarity,which determine the irreplaceable application value of fractal in various fields like physics,material science,geology,mathematics,and biology.Fractal and chaos,as important branches of the nonlinear science,are bound tightly.Chaotic attractor is also a fractal set,which is the initial point set of those unstable trajectories in the dynamical system.Although the studies of chaos and fractals have long been mature,the fractal processes and chaotic systems are seldom combined to generate more abundant chaotic attractors.Compared with the single-scroll chaotic attractors,the multi-scroll chaotic attractors have much higher complexity and better adjustability,which make multi-scroll chaotic attractors have wider application prospects in chaos-based information technologies,such as encryption and secure communication.Therefore,it is an attractive but challenging task to create a hyperchaotic attractor with multi-scrolls.More importantly,the traditional methods of generating multi-scroll chaotic attractors,such as piecewise linear functions,switching manifolds,step functions and saturation sequences,make the chaotic system rough.In this paper,the fractal process is applied to the chaotic system to generate a multi-scroll chaotic attractor,which makes up for the shortcomings of the traditional methods.This paper thoroughly studies the combination of fractal and chaotic systems and the generation of multi-scroll chaotic attractors.Firstly,the mapping relation is obtained based on the Julia fractal expression.The mapping is further applied to a known chaotic system based on flux-controlled memristor to generate a new multi-scroll chaotic system.Moreover,a new method for generating multi-scroll chaotic attractors is explored.Secondly,a three-dimensional chaotic system based on flux-controlled memristor is proposed.The basic dynamic behaviors of the new chaotic system are investigated through symmetry,dissipation,stabilization of equilibrium points,Lyapunov exponent spectrum,power spectrum and Poincaré map.The mapping relations of Julia fractal,deformable Julia fractal with coefficients,high-order Julia fractal and polynomial Julia fractal are applied to the system,and abundant multi-scroll chaotic attractors are obtained.Also,the influence of a complex parameter on the system is analyzed.Thirdly,a four-dimensional hyperchaotic system based on flux-cntrolled memristor is established to analyze the dynamic characteristics of the system,such as Lyapunov exponent spectrum(system with two positive Lyapunov exponents),chaotic attractor,symmetry and dissipation,time-domain waveform of state variables,initial sensitivity and power spectrum.Moreover,one fractal process and two fractal processes are introduced to the state variables of the hyperchaotic system,both of which can produce circular multi-scroll chaotic attractors.Finally,based on the above research,two methods for generating multi-scroll chaotic attractors by the fractal process are deduced and applied to the classic Lorenz system,Chen system and Lü system.The numerical simulation results prove the effectiveness and feasibility of the proposed method.The research provides new methods and new ideas for the design of multi-scroll chaotic systems.