Bound, Bifurcation, Control And Synchronization For Some Classes Of Chaotic Systems

Chaos is an existing phenomenon in nonlinear systems, which arises from deterministic system in a random and non-reguar fashion. It is ubiquitous and complex. The most reamarkable chaotic character is that chaotic systems exhibit high sensitivity to initial conditions. In other words, a nonlinear system called chaotic system is unpredictable. Chao breaks the rigorous division between deterministic systems and random systems, which is un-repeatable, local un-stable and whole stable. In the study on chaos, The bound of chaotic systems have become front research topics, extremely challenging. The ultimate bound of a chaotic system is important for the investigation of the qualitative behavior of a chaotic system, however, estimating the globally exponentially attractive and ultimate bound set for a dynamic system is a quite challenging task in general. A chaotic system is bounded, in the sense that its chaotic attractor is bounded in the phase space, and estimate of its bound is important in chaos control, chaos synchronization and applications. On the one hand, Hopf bifurcation of a hyperchaotic system is studied in this dissertation, on the other hand, bound, control, synchronization and some issues concerned of several novel chaotic systems are investigated in this paper.The main content is divided into the following aspects:1. The Hopf bifurcation of a new modified hyperchaotic Lii system is investigated. Firstly, A detailed set of conditions are derived, which guarantee the existence of the Hopf bifurcation. Furthermore, by applying the normal form theory, the direction and type of the Hopf bifurcation, and the approximate expressions of bifurcating periodic solutions and their periods are determined. In addition, numerical simulation results supporting the theoretical analysis are given.2. Two novel synchronization criterions of the unified chaotic systems with known or unknown parameters is investigated. Based on LaSalle's invariance principle and linear matrix inequality (LMI) formulation, firstly, a simple linear feedback controller with the updated gain is obtain to make the state of two identical unified chaotic systems asymptotically synchronized, secondly, a simple adaptive controller is proposed for synchronization of the unified chaotic systems with unknown parameters. Finally, simulation results are given to verify the theoretical analysis.3. Based on the Shilnikov theorem, a new three-dimensional square chaotic system, which has tow equilibrium poits. There is hyperbolic saddle focus. The formation mechanism shows that this chaotic system has Smale horseshoes (homoclinic chaos). Numerical simulations are given to show the effectiveness of the theoretical results. Finally, Smale horseshoses has been found in the system using undetermined-coefficient method, and it is chaotic in the sense of Shilnikov.4. In this work, a dynamic outputs feedback controller for a class for a class of chaotic systems is developed for the first time. For stability, a well-known Lyapunov stability theorem combining with linear matrix inequality optimizaion is utilized. A numerical simulation is presented to show the feasibility of the proposed scheme.5. A robust adaptive control is proposed to realize chaos synchronization and parameters identification between two different chaotic systems with uncertainties, bounded time-varying unknown parameters, and noise perturbation. Based on Lyapunov stability theory, a robust control law and a parameters identification scheme are presented. Finally, numerical esults are presented for the Lorenz, Chen systems. Numerical simulations are given to demonstrate the robustness and efficiency of the proposed method.6. A typical autonomous chaotic system is firstly presented in this paper which contain two smooth quadratic terms of systemic variables. When the parameters of system are changed, the typical system belongs to or does not belong to the generalized Lorenz system. Basic dynamical properties of the typical system are studied in terms of numerical simulation, Lyapunov exponent spectrum, bifurcation diagrams and Poincare section diagrams. At last, the ultimate bound of typical system is estimated and the expression of the bound is derived for all the positive values of its parameters a,b,c,d.7. At first, under the certain condition, the theorem about an upper bound estimate of two variables was given by parameterization and proved. Secondly, the theorem about and upper bound estimate of three variables was proposed. Then, the condition of generating unstable saddle focus equilibrium points is presented. Therefore, a chaotic system was constructed.8. A novel chaotic system which contains two smooth biquadratic terms of systemic variables and three real equilibriums is presented. The characters of three equilibriums are discussed. The dynamic properties of the novel system are investigated via theoretical analysis, numerical simulation, Lyapunov dimension, Poincare diagrams, Lyapunov exponent spectrum and bifurcation diagrams. Finally, in the light of topological horseshoe map theory, the paper analyses the existence of topological horseshoe in the presented autonomous chaotic system.9. A chaotic system is presented and the dynamic properties of the novel system are investigated in term of Lyapunov dimension, Poincare diagrams, Lyapunov exponent spectrum and bifurcation diagrams. At last, the impulsive control of the system is global asymptotical stable at origin.