Jacobi Analysis Of Several Types Of Three-dimensional Systems

Kosambi-Cartan-Chern(KCC)theory is a geometrodynamical approach,which was put forwarded based on the D.D.Kosambi,E.Cartan and S.S.Chern's pioneering works in the1930 s.It has been widely applied in biology,chemistry,physics and other fields.In this dissertation,the Jacobi stability of trajectories of three types of first-order differential equations are studied respectively by KCC theory.Then the chaotic mechanisms of systems are explored.The main contents are as follows.In Chapter 1,the research background,significance and status quo of this paper are expounded.Some basic concepts and conclusions of the Lyapunov stability and KCC theory are briefly introduced.The relationship between linear stability and Jacobi stability of twodimensional systems is summarized.The concepts of instability exponent and curvature of the trajectory of the system are introduced.In Chapter 2,according to classification of the Jordan canonical form of the linear matrix of the system,the Jacobi stability of equilibrium points of seven types of three-dimensional linear systems are analyzed respectively.The obtaining results show that the equilibrium points of seven types of three-dimensional linear systems are always Jacobi unstable.In Chapter 3,the Jacobi stability of trajectories of a second-order differential system is considered,based on the classical three-dimensional Chen system.By calculating the five geometric invariants of the system,Jacobi stable conditions for the three equilibrium points of the system are obtained,and the obtaining results are compared with those of Lorenz system.It is found that a Lyapunov stable periodic solution of Chen system is Jacobi unstable.By introducing instability exponent and curvature,the chaotic mechanism of the system is analyzed tentatively in combination with numerical simulation.In Chapter 4,the Jacobi stability of trajectories of a differential system,which is composed of three second-order differential equations,is considered based on the three-dimensional Rabinovich system.And then five geometric invariants of the system are obtained to analyze the Jacobi stability of the equilibrium points.In addition,the chaotic mechanism of the system is analyzed tentatively by combining numerical simulation.In Chapter 5,summary is drawn,and some future research works are put forward.