Study On Complex Dynamics Of Controlled High-dimensional Hyperchaotic Systems

In 1963,Lorenz simplified a 3D quadratic polynomial system from weather forecasting models and found a very interesting chaotic attractor.Since then,chaos has become a global research focus in the nonlinear field.Chaotic systems have also been widely used in engineering.In 1979,R¨ossler proposed a hyperchaotic system with two positive Lyapunov exponents.Such system has more complex dynamic behavior than the general chaotic system and has been widely studied.This leads to a deeper understanding of chaos and allows chaos to be used better.However,it is difficult to predict the dynamic behavior of chaotic and hyperchaotic systems because the solutions cannot be obtained.The dynamic properties of high-dimensional systems are particularly complex and the study is more difficult.So far,there is no unified method for the design and study of chaotic and hyperchaotic systems.Chaos and hyperchaotic systems can be classified according to the number and stability of the equilibrium points.A system with no equilibrium point or only stable equilibrium point is a hidden system.The basin of attraction of the self-excited attractor intersects with the unstable manifold of equilibria.Self-excited systems have been studied a lot,but hidden chaotic systems have been little studied.In high-dimensional systems,it is difficult to study both self-excited and hidden chaotic systems,especially those with hidden and hyperchaotic characteristics.At present,the study of hyperchaos mainly focuses on the four-dimensional case,few results of high-dimensional hyperchaotic systems have been obtained.There is no suitable theory and method to study chaos systematically.Based on a class of 3D chaotic systems with stable equilibrium,several kinds of high-dimensional hyperchaotic systems are found,including a 5D hidden hyperchaotic system,a 6D hidden hyperchaotic system and a 7D hyperchaotic system.The dynamic properties of these systems were studied in detail by using phase diagram,Lyapunov exponent,bifurcation diagram and so on.Then these hyperchaotic systems are designed and the physical realization of the systems is carried out by circuit.Furthermore,the parameter switching algorithm is used to study the relationship between different types of attractors.This paper has four chapters,the specific content is as follows.Chapter 1 gives the background and significance.It briefly reviews the development,basic theories,characteristics and research methods of chaos.We also introduce several typical chaotic and hyperchaotic systems in the end.Chapter 2 proposed a new 5D hidden hyperchaotic system that has three positive Lyapunov exponents under no equilibrium and only one stable equilibrium.The phase portrait,bifurcation diagram,Lyapunov exponents spectrum and other methods are used to study the dynamic behavior of the 5D hyperchaotic system.The evolution of self-excited chaotic attractors,hidden chaotic attractors and hidden hyperchaotic attractors are studied.Moreover,parameter switching indicates that hidden attractors can be approximated by switching between self-excited attractors.Finally,circuit experiments show that the 5D hidden hyperchaotic system exists in the real world.In chapter 3,a new 6D hidden hyperchaotic systems is proposed.The system has hyperchaotic attractor with four positive Lyapunov exponents when there is no equilibrium.When the system has an equilibrium line and two equilibrium lines,two different kinds of singularly degenerate heteroclinic cycles can be found.Moreover,one can observed6 D chaos and hidden hyperchaos as the singularly degenerate heteroclinic cycles vanish with the perturbation of parameters.This indicates that the bifurcation of singularly degenerate heteroclinic cycle is a possible route to chaos.In addition,multistability can be found under three types of equilibria,especially the system has seven attractors coexisting under one equilibrium line.Moreover,the parameter switching algorithm is used to approximate 6D orbits.Finally,the circuit diagram is designed to physically carry out the 6D hidden hyperchaotic system,and the experimental results are consistent with the numerical results.In chapter 4,a novel 7D hyperchaotic systems with five positive Lyapunov exponents is proposed.Firstly,the dynamics evolution of the 7D system is studied in detail,and the results show that the dynamics of the system is very complex.What is particularly interesting is that singularly degenerate heteroclinic cycles can be obtained when the system has an equilibrium line.With the perturbation of the parameters,singularly degenerate heteroclinic cycles bifurcate to 7D chaotic and hyperchaotic attractors.Moreover,the method of parameter switching is used to approximate different types of 7D attractors.Finally,the physical implementation of the system is carried out with the circuit.