Complex Dynamics Of A New Four-dimensional Hyperchaotic System

Since Lorenz firstly found chaotic attractor at the deterministic system by numerical simulation.The study of some classical Lorenz type systems has set off a hot interest in the study of chaotic systems.Compared with 3D chaotic system,the 4D hyperchaotic system can exhibit more rich and complex dynamics under the structure and properties.In 1979,R¨ossler found the first 4D hyperchaotic system with only a single nonlinear term.Some important 4D hyperchaotic systems are further studied such as Chua circuit,Lorenz-Haken system,Lorenz-type system and so on.These chaotic or hyperchaotic systems have significant applications in biological neural network,nonlinear circuit,secure communications,image encryption,engineering and so on.The acting content of this paper is mainly as follows:In chapter 1,the development history of chaos research is introduced,namely the origin,research background and research significance of chaos.Some chaotic or hyperchaotic correlation theory is introduced.Some classical hyperchaotic systems are briefly introduced,and the research status of these systems are understanded.In chapter 2,this paper reports the complex dynamics of a four-dimensional(4D)hyperchaotic system with a unique or three isolated equilibria,a curve or two linear nonisolated equilibria.Based on the Lorenz type system,a 4D hyperchaotic system is obtained by adding feedback control and variable transformation.Furthermore,the existence and stability of equilibria are studied,The existence of hyperchaotic attractors near each equilibrium point is numerically simulated.Finally,varying parameter values are used to simulate the complex dynamical behavior of systems from periodic-hyperchaotic-chaotic.In chapter 3,the local dynamics of fork bifurcation,Hopf-bifurcation and zero Hopf-bifurcation are analyzed.Using center manifold theorem research fork bifurcation of the non-hyperbolic equilibrium and the stability of the non-hyperbolic equilibrium.By means of bifurcation theory,the stability of the limit cycle of Hopf bifurcation branch are analyzed and the results are verified by numerical simulation.Through the average theory to study the existence of zero Hopf-bifurcation,moreover,perturbation parameter values,one can obtain a path from periodic-quasiperiodic-chaotic-hyperchaotic.It is of great significance to study the structure and formation mechanism of chaos and hyperchaos.In chapter 4,this paper studies three different type of infinite many singular degenerate heteroclinic cycles,the coexistence of the singular degenerate heteroclinic cycles and the ultimate bounded of hyperchaotic attractor.What is more noteworthy is that the singular degenerate heteroclinic cycle can be distinguished into different types by the kind of equilibrium point and the different dimensions of the unstable manifold.By changing the value of the parameter,the singular degenerate heteroclinic cycle breaks up and degenerates into a saddle focus,moreover,complex dynamics such as period,chaos and hyperchaos appear.The ultimate boundness of the hyperchaotic system is proved by the method of constructing Lyapunov function and the optimization theory.Ultimately bound set not only an accurate description of the positions and boundaries,but also an important part to understand the structure of the attractors.In chapter 5,five different types of coexisting attractors are numerically simulated as follows: two periodic attractors,two quasi-periodic attractors,two chaotic attractors,periodic and quasi-periodic attractors,periodic and chaotic attractors.The coexisting attractor makes the system complex and multi-stable.The theoretical method for studying the coexisting attractor is not mature yet,but it is of great significance to demonstrate the complexity of the hyperchaotic system by numerical simulation.