Geometric Analysis Of Several Chaotic Systems Based On KCC Theory

Chaos refers to the seemingly random irregular motion in a deterministic system,and its behavior is uncertain,unrepeatable and unpredictable.It has been widely used in many fields such as secure communication,biological science,information technology and engineering technology,etc.Due to the application potential of chaos in engineering,more and more experts and scholars are stimulated to explore the intrinsic complexity mechanism of chaotic system from different perspectives.This dissertation starts from the perspective of differential geometry,based on the geometric invariants of KCC theory,Jacobi stability of the equilibrium points of three dimensional linear flow are studied,and the Jacobi stability and complexity of the trajectories of three dimensional Chen chaotic system,three dimensional Hindmarsh-Rose chaotic system and four dimensional R¨osser hyperchaotic system are analyzed.The chaos mechanism is further discussed.The main contents are as follows:In Chapter 1,the research background,significance and status quo of this paper are expounded.Some basic concepts,related properties and existing conclusions of the Lyapunov stability theory and KCC theory of differential equations are briefly described.The definitions of instability exponent and curvature of the trajectories of the system,and the basic concepts and notations related to Riemannian manifold are introduced.In Chapter 2,the Euler-lagrange extension theory and Jacobi stability theory of a flow on a Riemannian manifold are expounded,and the theories are applied to the study of the equilibrium solutions of three dimensional linear flow with seven Jordan standard types.Furthermore,and the Lyapunov stability and the Jacobi stability of the equilibrium points are compared and analyzed.In Chapter 3,the Chen system is transformed into a second order differential system controlled by the the Boltzmann-Hamel equation firstly,the geometric invariants of the system are obtained.Then,the relationship between the torsion tensor and the complex dynamic behavior of the system is discussed from the perspective of the tangent bundle.The results show that the chaotic behavior of the Chen system can be geometrically explained by the torsion tensor.Finally,the Chen system and the Lorenz system are compared and analyzed,it is found by the torsion tensor that the Chen system has more complex geometric dynamics phenomena than the Lorenz system under the specific parameter conditions.In Chapter 4,the Lyapunov stability of the equilibrium points of three dimensional Hindmarsh-Rose chaotic system is analyzed firstly.Then,based on the Euler-lagrange extension theory of a flow on a Riemannian manifold,the system is transformed into an equivalent system composed of three second order differential equations.By using the Jacobi stability theory of a flow on a Riemannian manifold,the Jacobi stability conditions of the equilibrium points of the equivalent system are obtained.By comparison,it is found that the Lyapunov stability and the Jacobi stability of the equilibrium points of the system are not completely consistent.By means of the time evolution of the deviation vector,its components and the instability exponent,the dynamic behaviors of the trajectories near the equilibrium points of the system are characterized.Finally,with the help of introducing the curvature of the deviation vector,the chaos generation mechanism of the system is analyzed in combination with numerical analysis results.In Chapter 5,the local bifurcation phenomenon of four dimensional R¨ossler system is analyzed firstly.Then,the system is transformed into an equivalent system composed of two second order differential equations.By calculating the deviation curvature tensor and its eigenvalue of the equivalent system,the Jacobi stability of the trajectories of the system(including equilibrium points and periodic orbit)is obtained.By comparing the Lyapunov stability and Jacobi stability of the equilibrium points of the system,it is found that the two types of stability are not always consistent.By means of the time evolution of the deviation vector,its components and the instability exponent,the dynamical behaviors of the trajectories near the equilibrium points of the system are described.Finally,the curvature of deviation vector is used to describe the trajectory behavior of four dimensional R¨ossler system near the equilibrium points,and the relationship between the change of curvature and the generation of chaos is explored.In Chapter 6,the summary of this study is given,some further research ideas are put forward.