Study Of Construction Of Time-delayed Chaotic System And Its Synchronization As Well As Simulation

In security communication, the cryptogram generator is a key device. It is showed that in chaotic security communication, this cryptogram generator can be a chaotic system. In chaotic security communication, the utility of chaotic system is "encryption", thus it is valuable to construct a proper chaotic system for chaotic security communication. While the duty of chaotic synchronization is "decryption", it is a key technology for chaotic security communication.It is showed by large amount of uptodate investigations of nondelay chaotic system that those low dimensional nondelay chaotic systems with few positive Lyapunov exponents were not safe enough. To increase the number of positive Lyapunov exponents, the dimension of a system should be increased drastically, and the structure of the synchronization system would be huge and complex, so it would be more difficult for synchronization. Time-delayed chaotic system is an infinite dimension system; it can be a hyperchaotic system with large numbers of positive Lyapunov exponents, and can generate highly stochastic and unpredictable time series. Thus in the process of synchronization, it overcomes the shortage of nondelay chaotic system with huge dimension and complex structure, and it can also increase safety for secure communication. Therefore the study of chaotic synchronization of a time-delayed chaotic system is of high practical importance for increasing the safety of secure communication. It may become a new increase point of new-higher technology. In addition, time-delayed chaotic system can be considered as a special case of spatio-temporal chaotic systems with extremely high safety for security communication. The study of synchronization of time-delayed chaotic system would be helpful for the study of synchronization for general spatio-temporal chaotic systems. Thus the study of synchronization of a time-delayed chaotic system is of theoretical significance.This thesis investigated two key problems for chaotic security communication, which is the construction of chaotic system ("cryptogram generator") and their synchronization ("decrypt"). By theoretical investigation as well as numerical simulation, some results with different aspect were established.Two methods were presented for constructing a time-delayed chaotic system: one is perturbation, time-delayed linear perturbation for chaotic system or time-delayed nonlinear perturbation for linear system, and the other is a time-delayed modification,adjusting the nonlinear term of a chaotic system to a delayed one. By amendment of stable linear system and nondelay chaotic system, two novel time-delayed chaotic systems with freedom degree 1 , six novel time-delayed chaotic systems with freedom degree 2 , five novel time-delayed chaotic systems with freedom degree 3 including a time-delayed feedback Chua circuit with freedom of degree 3 were constructed.An original definition of Lyapunov exponent for time-delayed system was presented, upon which a simple practical method for determining whether a time-delayed system was chaotic or not was presented: one first exams the wave form and the projection plot of attractor to find out whether a system is chaotic in the first round, then he draws the 'sensitive' wave form to strengthen the affirmed result of the first step, and at last he estimates the largest Lyapunov exponent to make an affirmative conclusion. This method was tested by several well known time-delay chaotic system and it was found to be reliable. The chaotic activities of new time-delayed system constructed were demonstrated by this method and it was found that all the new time-delayed systems were chaotic with certain parameters.For the investigation of synchronization of time-delayed chaotic systems, several asymptotic stability theorems with matrix inequalities as sufficient conditions for linear time-delayed system were proved by using stability theory of functional differential equations. Upon the analysis of the structure of the new time-delayed chaotic system constructed, two classes of abstract time-delayed chaotic systems were presented, one is the time-delayed nonlinear perturbation of a nondelay linear system and the other is nondelay or time-delayed nonlinear perturbation of a time-delayed linear system. These two classes of systems cover all the new time-delayed chaotic system constructed. The asymptotic stability theorems were used to investigate three kinds of synchronization problems (complete synchronization, lag synchronization and anticipating synchronization) of these two classes of time-delayed systems, and several sufficient conditions were established. And then the synchronization parameters were tested by these sufficient conditions for all new time-delayed chaotic system constructed, and the numerical simulation with Matlab 6.5 showed satisfaction.Adaptive synchronization and control problems for time-delayed system with parameters not fully known were investigated. Firstly, the adaptive chaotic synchronization problem of the time-delayed linear perturbed parameters fully unknown Chen system was studied by using Lyapunov-Krassovskii functional method, a parameters update law was established and the complete synchronization with thetime-delayed linear perturbed Chen system as drive system was reached. Then the adaptive chaotic synchronization problem of the time-delayed feedback Chua system with two parameters unknown was studied by using Lyapunov function method, a parameters update law was established and the complete synchronization with time-delayed feedback Chua system as drive system were reached. Finally, the Lyapunov-Krassovskii functional method was used to study the adaptive synchronization and control problems for a general class of time-delayed system, two general theorem were proved and a new time-delayed chaotic system constructed were used to simulat it.