Research For Generalized Projective Synchronization Of Fractional-order Chaotic Systems And Its Application

Chaos exists widely in nature with the complex nonlinear dynamic form. Since it has many excellentproperties such as the extreme sensitivity to initial values and parameters, non-periodicity, continuous andbroadband spectrum, noise-like, long-term unpredictability and so forth, chaotic systems are usuallyapplied in secure communication, cryptography and other information security fields, and have attractedgreat attentions from many researchers. In fact, the real chaotic systems are ideally generalized as theinteger-order chaotic systems. However, the dynamical behaviors can be more accurately described by thefractional calculus operators. Compared with integer-order chaotic systems, the fractional-order chaoticsystems not only have the characteristics of the integer-order chaotic systems, but its dynamical propertiesare closely related with the fractional order and the historical memory. Thus, fractional-order chaoticsystems have widely application prospects in information security, automatic control, image processing andother engineering fields.So far, in contrast to the integer-order chaos theory, the fractional-order chaos theory is a developingfield and imperfect. There exist many problems to be resolved. For example, the stability theory of thefractional-order chaotic systems is imperfect, some controlling or synchronization methods for theinteger-order chaotic systems can not be directly applied to fractional-order chaotic systems, and there arefew studies on synchronization and parameter identification of uncertain fractional-order chaotic systems,etc. In addition, the researchers have tried to apply the fractional-order chaos in cryptography,telecommunications, finance and other fields, and achieved some preliminary results. Therefore, theresearch on synchronization of the fractional-order chaotic systems has important theoretical significanceand great practical application value.According to the status quo of the research on the factional-order chaos, based on the fractionalcalculus theory and the prediction-correction algorithm, this paper analyzes the dynamic behaviors offractional-order chaotic systems, and studies the generalized projective synchronization betweenfractional-order chaotic systems with different structures, coupled structures or uncertain parameters, andits application in secure communication. The main content is as follows: Firstly, generalized projective synchronization problem between two fractional-order chaotic systemswith different structures is studied. Based on the stability theory of the fractional-order systems, thenonlinear controller is designed to achieve generalized projective synchronization between thefractional-order Lorenz system and the fractional-order Chen system. Theoretical analysis and numericalsimulations are provided to verify the validity of the synchronization scheme.Secondly, we present a new fractional-order chaotic system, analyze its dynamics behaviors, andpropose a generalized projective synchronization scheme of the coupled fractional-order chaotic systems.By introducing the fractional differential operator, a new fractional-order chaotic system is obtained. Theresults show that the system is still chaotic when the fractional order is greater than or equal to2.946. Thena coupled generalized projective synchronization scheme for the fractional-order chaotic systems ispresented. Finally, we design a fraction-order chaotic secure communication scheme by means of thechaotic masking technology. Numerical example further verifies the proposed fractional-order chaoticsynchronization method is effective and secure communication scheme is feasible.Thirdly, we study the adaptive generalized projective synchronization between fractional-orderhyperchaotic systems with unknown parameters and its application in secure communication. Based on theprediction-correction algorithm and computer simulations, the dynamical behavior of a newfour-dimensional fractional-order system is studied. The results show that the system is hyperchaotic whenthe minimum order of this system is3.128. Furthermore, an adaptive controller and the parameters updaterules are designed to obtain generalized projective synchronization of the fractional-order hyperchaoticsystems and identify the unknown system parameters simultaneously. Theoretical proof and numericalsimulations show the effectiveness of the proposed synchronization scheme. Finally, a chaotic securecommunication scheme based on adaptive generalized projective synchronization and the chaotic parametermodulation technology is presented. The simulation results demonstrate the feasibility of this scheme.