Complex Dynamics Of 6-Dimensional Hyperchaotic Systems Based On Lorenz System

Chaos theory is a significant branch in applied mathematics due to its great potentials in various theoretical and applications research fields in engineering, physics, economics,biology and other sciences. Chaos theory is concerned with the studying of the behavior of dynamical systems that are extremely sensitive to initial conditions. Lorenz system was discovered by Edward Norton Lorenz, who introduced basic contributions in the modern history of chaos. The research of Lorenz type systems follows the great improving process of chaos sciences. Also, hyperchaotic system is classified as a chaotic system with more complex dynamical behavior than ordinary chaotic system, therefore hyperchaotic synchronization has been a popular research area in many fields of applications.Based on Lorenz system, this dissertation introduces two new 6-dimensional hyperchaotic systems, and deeply explores the dynamical behaviors of these systems. To investigate the new systems more, the corresponding hyperchaotic and chaotic attractor are numerically verified through exploring phase trajectories, Lyapunov exponents, bifurcation, analysis of power spectrum and Poincar′e projections. Moreover, some complex dynamical behaviors such as the stability of hyperbolic equilibrium and many complete mathematical characterizations for 6D Hopf bifurcation are further rigorously derived and studied. When the parameters are known in advance, the one-way linear coupling approach to synchronize the 6D hyperchaotic system up to a scaling factor is applied and when the parameters are fully unknown, utilizing the adaptive method to synchronize the uncertain 6D hyperchaotic system up to a scaling factor with carrying out numerous numerical simulations to verify the validation and e?ciency of the proposed schemes.The main concept of this works are as follows:In Chapter 1, the chaos theory background and significance are defined. The historical background of chaos theory and Lorenz system, accomplishment and basic facts of chaos theory are introduced. The classical Lorenz system, Lorenz-type hyperchaotic systems and several other typical hyperchaotic systems are presented.In Chapter 2, a six-dimensional hyperchaotic system based on Lorenz system with four positive Lyapunov exponents under unique equilibrium is introduced, this is obtained by coupling between a 1D linear system and a 5D hyperchaotic system that is formed by adding a linear feedback controller and a nonlinear feedback controller to the Lorenz system. To further analyze the new system, the corresponding hyperchaotic and chaotic attractor are firstly, numerically verified through investigating phase trajectories,Lyapunov exponents, bifurcation, analysis of power spectrum and Poincar′e projections.Moreover, some complex dynamical behaviors such as the stability of hyperbolic equilibrium and two complete mathematical characterizations for 6D Hopf bifurcation are rigorously derived and studied.In Chapter 3, another new six-dimensional hyperchaotic system under unique equilibrium with four positive Lyapunov exponents or three equilibria with three positive Lyapunov exponents is introduced. Moreover, Lyapunov exponents, fractal dimension,chaotic and hyperchaotic behaviors are further numerically verified. The dynamical behaviors of this 6D hyperchaotic system such as the stability of hyperbolic equilibrium are further theoretically analyzed. Moreover, the existence of Hopf bifurcation have been proved.In Chapter 4, focus on the scaling synchronization issue. When the parameters are known in advance, applying the one-way linear coupling approach to synchronize the 6D hyperchaotic system up to a scaling factor. When the parameters are fully unknown,utilizing the adaptive method to synchronize the uncertain 6D hyperchaotic system up to a scaling factor. Finally, many numerical simulations to verify the validation and e?ciency of the proposed schemes were carried out.