Qualitative Theory And Synchronization In Complex Dynamical System

In recent years, with the rise of nonlinear science and application of computer technology, complex dynamical system theory is booming, involving multiple dis-ciplines of hyperbolic geometry, fractal geometry, the modern analysis and chaos. The chaotic system with complex variables is widely used in secure communication, electromagnetic wave amplitude, laser control, and many other fields. Therefore, this thesis focuses on the qualitative theory of complex dynamic system and a series of basic researches on its complex synchronization, mainly for the integer-order chaotic (hyperchaotic) complex dynamic systems with complex parameters and fractional-order chaotic (hyperchaotic) complex dynamic system. The main work and innova-tion points are as follows:1. A novel four-wing hyperchaotic complex system and its complex modified hybrid projective synchronization with different dimensionsFirstly, we introduce a new Dadras system with complex variables which can exhibit both four-wing hyperchaotic and chaotic attractors, and investigate its qual-itative theory and chaotic properties. Secondly, we present the definition of modi-fied hybrid project synchronization with complex transformation matrix (CMHPS). The complex transformation matrix is not square matrix, and its elements are com-plex numbers. By employing nonlinear control technique, CMHPS for different dimensional hyperchaotic and chaotic complex systems with complex parameters is achieved by complex feedback gain matrix and appropriate controller which is de-signed for a desired complex transformation matrix. Furthermore, CMHPS between the novel hyperchaotic Dadras complex system and other two different dimensional complex chaotic systems is provided as an example to discuss increased order syn-chronization and reduced order synchronization, respectively. It is the first time to achieve the complex synchronization for chaotic complex systems with complex parameters without separating real and imaginary parts of complex parameters or complex variables.2. Complex modified projective synchronization of chaotic (hyperchaotic) com-plex dynamic systems with complex parametersWe propose modified projective synchronization with complex scaling matrix (CMPS) for a class of complex chaotic (hyperchaotic) systems with complex pa-rameters. The scaling matrix is a diagonal matrix, and its elements are complex numbers. Since complete synchronization (CS), anti-synchronization (AS), projec-tive synchronization (PS) and modified projective synchronization (MPS) are special cases of CMPS, and real parameter is extreme case if the imaginary part of complex parameter is zero, CMPS for complex systems with complex parameters contains existing works and extend previous works. It is necessary to point out that, unlike the schemes proposed in the literature, we do not separate the real and imaginary parts of the complex variables or complex parameters. By choosing appropriate Lya-punov functions dependent on complex variables, and employing nonlinear control technique and adaptive control technique, sufficient criteria on ACMPS are derived. Moreover, in the complex space, the slave system can be asymptotically synchro-nized up to the projection of nonidentical or identical master system by a desired complex scaling matrix, and all of unknown parameters in both master and slave systems are achieved to be identified by virtue of the complex update laws. Numeri-cal results verify the feasibility and effectiveness of the presented schemes.3. Complex modified function projective synchronization of chaotic complex dynamic systems with complex parametersWe put forward modified function projective synchronization with complex scaling function matrix (CMFPS) for a drive-response complex system with com-plex parameters. The scaling function matrix is a diagonal matrix, and its elements are complex functions. In particular, by constructing appropriate Lyapunov func-tions dependent on complex variables, and employing adaptive control technique, sufficient criteria on adaptive CMFPS and parameter identification are derived. Fur-thermore, in the complex space, the response system can be asymptotically syn- chronized up to the projection of nonidentical or identical drive system by desired complex scaling functions. The unknown complex parameters converge to their true values based on the linear independence. The CMFPS scheme can be applied to the practical system more widely, and further enhances the effect of secrecy.4. Complex modified generalized projective synchronization of fractional-order chaotic complex systemsFirstly, we introduce a new fractional-order modified Lu systems, and inves-tigate its qualitative theory and chaotic properties. Secondly, we present modified generalized projective synchronization (MGPS) with a transformation matrix for fractional-order chaotic systems. The transformation matrix is non-diagonal square matrix, and its elements are real numbers. Based on the stability theory of fractional-order systems, by employing nonlinear feedback control technique, MGPS between the new system and the fractional-order hyperchaotic Lorenz system is implemented as an example. Thirdly, the transformation matrix is extended to complex matrix, and the real fractional-order system is extended to the complex fractional-order system, we put forward modified generalized projective synchronization with complex trans-formation matrix (CMGPS) between fractional-order complex chaos and real chaos and between two fractional-order chaotic complex system, respectively. Three nu-merical examples verify the feasibility and effectiveness of the presented schemes. It is the first time to achieve CMGPS between fractional-order complex chaos and real chaos and between two nonidentical fractional-order chaotic complex system, respectively.To sum up, this dissertation focuses on the qualitative theory of complex dy-namic system and a series of basic researches on its complex synchronization, and has improved Lyapunov stability theory for complex dynamic systems. All kinds of complex synchronization of chaotic complex systems were achieved, providing theoretical basis for further strengthening the security of communications.