Generation, Control And Synchronization Of Several Types Of Chaotic Systems With Complex Topological Structure

Ring multi-scroll, nesting multi-wing, grid multi-scroll and multi-torus, multi-folded torus, high-dimension generalized cat map and so on in phase space are called chaotic attractors with complex topological structure. Compared with double scroll or double wing attractors with simple topological structure, the above-mentioned chaotic attractors with complex topological structure have very complex topological structure and more complicated dynamical behavior. Generation, control, synchronization and some issues concerned of several new types of chaotic attractors with complex topological structure are studied in this dissertation. The main content is devided into the following aspects:1. As the study on chaos becomes more in-depth, some new multi-scroll chaotic and multi-wing chaotic systems arises one after the other. In recent years, these systems become very active research focuses in nonlinear circuits and systems. First, generation of several typical chaotic attractors with complex topological structure are studied in this dissertation. The following three aspects are finished:(1) A new multi-scroll hyperchaotic Chua system is proposedAlthough many approaches for generating multi-scroll chaotic attractors from 3-D Chua system have been intensively studied, there are few detailed reports regarding the multi-scroll hyperchaotic attractors from 4-D Chua system up to now. In addition,4-D Hyperchaotic Chua system can be constructed so far by some different techniques, such as two or more 3-D coupled Chua circuits, expansion ofπtype circuit configuration, both decomposition-based and junction field-effect transistor-based approaches, state feedback control and so on. It is believed that state feedback control-based approach has some advantages for generating various 4-D hyperchaotic systems from many existing 3-D chaotic systems. Moving forward from the above-mentioned accomplished works, in this dissertation, a 4-D hyperchaotic Chua system via state feedback control is constructed, by introducing with both piecewise-linear nonlinearity and smooth and piecewise smooth cubic nonlinearity. Furthermore, dynamical behaviors of this hyperchaotic system are further investigated, including Lyapunov exponents spectrum. bifurcation diagram and solution of state equations. In addition, a circuit is designed for 4-D hyperchaotic Chua system such that the double-scroll and 3-scroll hyperchaotic attractors can be physically obtained.(2) A new multi-wing chaotic system is proposedThe controller obtained by the improved C-A synchronization is modified by computer simulations to generate a special multi-wing chaotic attractor. By the theoretical analysis and numerical simulations, some basic dynamical properties of the controlled Lorenz system are investigated. Research results show that this system has complex and interesting chaotic and hyperchaotic dynamics.(3) Two four-dimensional fractional-order multi-scroll hyperchaotic Chua systems are proposedTwo four-dimensional fractional-order multi-scroll hyperchaotic Chua systems are gained based on the four-dimensional integer-order multi-scroll hyperchaotic Chua systems, including a four-dimensional fractional-order hyperchaotic Chua system with both smooth and piecewise smooth cubic nonlinearity, and a four-dimensional hyperchaotic Chua system with piecewise-linear nonlinearity. By utilizing the fractional calculus theory and computer simulations, it is found that multi-scroll hyperchaotic attractor exists in this fractional-order Chua system with order less than 4.2. On study of control and synchronization of chaotic attractors with complex topological structure, the following four aspects are finished in this dissertation:(1) A fractional order hybrid feedback controller is designed to synchronize the multi-scroll Chua systemThe fractional controller converts the system behavior with integer derivatives into a system with fractional derivatives. This increases the degree of freedom of the system by means of providing the stability. In addition, an integer hybrid feedback controller is used to decrease the time of synchronization. The proposed controller makes use of the good of both integer and fractional order controllers. The performance of the controller is shown via numerical simulation when it is used to synchronize the multi-scroll Chua system.(2) An improved C-A projective synchronization scheme is proposed The complete synchronization and anti-synchronization(C-A synchronization) can be achieved when chaotic system has some degree of symmetry. In this paper, C-A synchronization is improved based on the linear separation method to achieve the C-A projective synchronization, that is, the drive and response vectors synchronize up to the scaling factor in the Lorenz system.(3) A unified adaptive controller and parameters update law are designed for achieved the adaptive full state hybrid function projective lag synchronization (FSHFPLS) in chaotic continuous-time systemFor the projective synchronization, most of research efforts have concentrated on studying the constant scaling factor. Based on the Lyapunov stability theory and FSHPS, this scheme is investigated in chaotic continuous-time system, and a unified adaptive controller and parameters update law are designed for achieved the projective lag synchronization up to a desired scaling function. In addition, a scheme for secure communication is presented. Numerical simulations are performed to verify and illustrate the analytical results.(4) An improved generalized Hamiltonian systems approach is proposedBy using the Lyapunov stability theory and some matrix techniques, a new sufficient criterion, formulated in the LMI form, is established. The new sufficient criterion can guarantee that chaotic synchronization is achieved at an exponential convergence rate. Then this improved approach is used to synchronize Lu system and n-scroll chaotic attractors in a modified Chua's circuit with hyperbolic tangent functions.Besides, synchronization between the fractional-order multi-scroll hyperchaotic Chua systems is achieved by a fractional active sliding mode controller and a suitable linear controller applied to the response system.3. Particle Swarm Optimization (PSO) which is an evolutionary computation is studied in this dissertation. Its research and application are not interrupted since it was put forward. In this dissertation, there are some problems of the randomness of initialization process and easy to fall into a local optimal solution when using PSO algorithm. In this paper, PSO algorithm is improved. The basic idea is:at the beginning, a great deal of initial population are produced by ergodicity of chaos, and then the initial particle swarm is preferred out; during the running time of PSO algorithm, to adjust the current particle by chaos so that the solution jumps out local convergence. In the end,Chaos PSO algorithm is proposed to optimize the integrated GM(1,1) parameter optimization model.This work was supported by the National Natural Science Foundation of China under Grant Nos.60871025 and 61172023, the Specialized Research Foundation of Doctoral Subject Point of Education Ministry under Grant No.20114420110003, the Natural Science Foundation of Guangdong Province under Grant Nos. 8151009001000060 and S2011010001018, and the Science and Technology Program of Guangdong Province under Grant No.2009B010800037.