Construction And Dynamics Analysis Of Chaotic Systems Using Hamiltonian Method

Chaotic dynamics is a very important branch of nonlinear science.Chaos theory can help explain complex nonlinear phenomena widely existing in the fields of mathematics,physics,chemistry,optoelectronics,economics and engineering science.With the discovery of the potential value of chaos in engineering application,the theoretical research on chaos has been enriched and deepened constantly.In the last decade,the construction and theoretical analysis of chaotic systems have become a hot spot for the study on chaos theory,but there are few systematic methods for constructing chaotic systems.Most of physical systems are open systems with energy storage,energy dissipation,energy generation and energy exchange with the external environment.Generally,open systems can be represented by generalized Hamiltonian systems,the key of which is the Hamiltonian energy function.The Hamiltonian energy includes the system's kinetic energy and potential energy.Based on the Hamiltonian energy theory,an open system can be decomposed into the system with different force fields.This thesis studies the general structure of continuous autonomous chaotic systems and the effects of different types of force fields on the system dynamics.The main problems to be addressed include:To begin with,a novel method for constructing different types of chaotic systems based on Hamiltonian energy function is proposed.By configuring the conservative force field,dissipative force field and external force field of these systems,several chaotic systems are obtained,including two three-dimensional dissipative systems,two three-dimensional non-Hamiltonian systems without conservation of energy,and three high-dimensional non-Hamiltonian systems with conservation of energy.Among them,the dissipative systems can generate chaotic motion with Lyapunov dimension closer to THREE,the non-Hamiltonian systems without conservation of energy have the characteristics similar to the Nosé-Hoove system,and the high-dimensional non-Hamiltonian system with energy conservation can generate chaotic and hyperchaotic motions on the isosurfaces subjected to their Hamiltonian.Then,two Hamiltonian energy functions,which correspond to the Lorenz system and the Chen system,respectively,are obtained from the unified Lorenz system based on the Hamiltonian energy theory.According to the geometric structure of the obtained Hamiltonian in phase space,it is proved that the Chen system is not a special case of the Lorenz system.Thirdly,the index theorem and its corollary in two-dimensional nonlinear systems are studied,and a method for constructing complex multi-wing chaotic systems is proposed.In particular,the mechanism of the complex dynamics of the eight-wing chaotic system is analyzed by using the Hamiltonian energy theory.Besides,the linear system theory can be applied in the analysis of a class of nonlinear systems and the simplest memristive system with multi-wing chaotic attractors is proposed.By presetting the motion of state variables in different regions on the phase plane to configure the state function and internal characteristic function of a memristor,the complex multi-wing chaotic dynamics are obtained.Moreover,through the Hamiltonian energy theory,the different force fields contained in the memristive system are investigated.