Chaotic Control, Synchronization And Applications In Information Encryption

Nonlinear science is a foundational discipline which concerns the common properties of nonlinear phenomena. Chaos theory and its application are one important subdiscipline of nonlinear science. In this paper, some problems about chaotic control, chaotic synchronization and its application in cryptography are explored and investigated. The main achievements in the research can be summarized as follows.Firstly, chattering problem in traditional sliding mode control for uncertain chaotic system is deeply studied. Based on variable universe theory, a variable universe adaptive fuzzy control is designed to replace the discontinuous sign function of the reaching law in traditional sliding mode control. By adjusting scaling gains, the fuzzy universe is changed automatically and the rules are dynamically adjusted. Thus, control accuracy is greatly enhanced, and the difficulty of design fuzzy rules is greatly reduced. The proposed variable universe fuzzy slide mode controller can control the chaotic system to a desired state in the presence of uncertainties and disturbance. Moreover, the higher chattering phenomenon in traditional sliding mode control does not occur.Secondly, the projective synchronization scheme of different chaotic systems with nonlinearity inputs is proposed. Starting form the practical application aspect, we consider that in the practical applications, owing to physical limitations, there always exist system uncertainties, external noise disturbances and nonlinear effects in the control actuators. Based on adaptive technique and sliding mode control method, an adaptive projective synchronization scheme is designed to ensure different chaotic systems with nonlinearity inputs can be rapidly synchronize up to the given scaling factor. Furthermore, the designed controller is robust to the parameter uncertainties and noise disturbance, without requiring the bounds of the parameter uncertainties and noise disturbance. In addition, the proposed control method can be further extended to the projective synchronization of different chaotic systems with more state variables.Thirdly, the dynamic behaviors of a new chaotic system and its generalized projective synchronization are studied. Based on the stability theory of fractional order systems, the dynamic behaviors of a new chaotic system in different fractional order are investigated. Firstly, we theoretically analyze and compute the order range in which the new fractional order system is chaotic. Then, the effectiveness of our analysis results is further verified by numerical simulations and computing the largest Lyapunov exponent. Furthermore, a nonlinear feedback controller is design to achieve the generalized projective synchronization of the fractional order chaotic system, and its validity is proved by Laplace transformation theory.Fourthly, the method of generating hyperchaotic system from a three-dimensional chaotic system is investigated. A new hyperchaotic system is presented by adding a nonlinear controller to the three-dimensional autonomous chaotic system. The generated hyperchaotic system undergoes hyperchaos, chaos, and some different periodic orbits with control parameters changed. The complex dynamic behaviors are verified by means of Lyapunov exponent spectrum, bifurcation analysis, phase portraits and circuit realization. The results of the hyperchaotic circuit implement were well agreed with the simulation results. For this three-dimensional autonomous chaotic system, we give some discussions on how to build the control law so that hyperchaos can be observed.Fifthly, the anonymous key agreement protocol proposed by Tseng et al. is analyzed and a new and more secure protocol is proposed. Firstly, it is demonstrated that Tseng et al.’s protocol can not guarantee user anonymity and protocol security against an insider adversary who is a legal user, and it can not provide perfect forward secrecy. Then, in order to conquer these problems, a new key agreement protocol is presented based on Chebyshev chaotic maps. In contrast with Tseng et al.’s protocol, the proposed protocol is more secure and preserves user anonymity.This research is supported by the National Natural Science Foundation of China (No: 60573172,60973152), the Superior University doctor subject special scientific research foundation of China (No:20070141014) and the National Natural Science Foundation of Liaoning province (No:20082165).