The Research On The Fractional Chaos Synchronization And Control

Chaotic systems always exist in nonlinear dynamic systems, and have been well studied in many fields such as physics, chemistry, biology and information. It's especially suitable for information encryption, because of some features that chaotic system possesses such as high randomicity, board spectra and highly sensitivity to initial conditions. Fractional calculus is a generalization of integration and differentiation to noninteger order fundamental operator. It has been found that many systems can be described by fractional differential equations, such as electromagnetic wave and other fields. Considering the fact that the dynamic behaviours are closely related to the system order and the historical memory effect for fractional operator, the dynamic behaviours created by fractional order system are more complex than that of tranditional integer order chaotic system. It has a great potential application in chaotic secure communication. Recently, the study of fractional chaotic system has received a growing attention since the rapid development of the computer engineering. This paper studies the fractional chaos control and synchronization problem for a kind of fractional chaotic system, and makes a consideration on the engineering application such as secure communication.Firstly, some special problems have been studied such as the numerical solution. Based on the frequency response of the fractional integral operator, the limitation of frequency domain approximation is well studied. The effectiveness is verified by the simulation of Chen system. Meanwhile, considering the fact that chaos is a new term in science to many people, this paper makes a survey of some basic concepts for chaos, the methods in chaos control and chaos synchronization, as well as the definition, character, stability and numerical algorithms of the fractional system. In the survey, we try to make a conparation between the methods for fractional order chaotic and that of integer order system to introduce some background knowledge for the following sections.Secondly, this paper proposed two methods for fractional order multi-scroll chaotic attractor generation. To replace the piecewise-linear function of the traditional Chua's circuit by smooth sine-function, the first multi-scroll chaotic attractor generation method is proposed. The second multi-scroll chaotic attractor generation method is based on the fractional swithed system. Furthermore, based on the stability of the fractional system and the Lyapunov exponents, the lowest system order for the multi-scroll chaotic attractor existence is studied. Theoretical analysis and numerical examples are provided to verify the effectiveness of the proposed methods.Thirdly, this paper proposed two methods for fractional chaotic system control. The first method is an improved version of PIλDμcontroller. This controller has only two control parameters and can be designed very easily. However, there are some limitations for the first method because the fractional chaotic system can't be stabilized to desired equilibrium point. To overcome this limitation, the other method is proposed. The second method is implemented with an increased system dimension. By introducing new variables and feedback into original system, the new system is created which have the same equilibrium points and dynamic behaviours in the neighborhood of those equilibrium points. The original fractional chaotic system can be stabilized into desired equilibrium points by stabilizing the new system to its equilibrium points, correspondingly. Numerical examples based on fractional Liu system and multi-scrolls are provided to verify our results.Fourthly, this paper proposed two methods for fractional chaotic synchronization which are adaptive synchronization and the method based on the Extended Fractional Kalman Filter (EFKF). Those two methods, which may be familiar for us, are theoretical basis for the secure communication. This paper focuses on those two synchronization methods and theirs generalization form in fractional system. The main contribution of this paper is to present some significant result for unknown parameter identification which is so important for the chaotic parameter modulation in secure communication. Based on the adaptive control, the first method can be applied to integer order system and the fractional system with the system order less than one. The proposed adaptive synchronization method can successfully synchronize two different structure fractional chaotic systems and identify the unknown parameters. The second method based on EFKF can be viewed as an observer problem. The EFKF synchronization method is an optimal algorithm for recursive estimation of the states and the unknown parameters. For the discrete fractional order chaotic system with unknown parameters, synchronization can be realized by the proposed two methods. Meanwhile, the unknown parameter identification would be successful if the column vectors of coefficient matrix for the unknown parameters are linearly independent. Taking the classical fractional system as numerical example, a series of examples are presented to verify the effectiveness of our conclusion.To end of this paper, we introduced a secure communication scheme which is based on chaotic parameter modulation. The proposed method can overcome some limitation of the chaotic masking. Based on the EFKF, in the presence of channel noise and processing noise, the proposed parameter modulation scheme can successfully transmit the binary digital signal with suitable processing in the receiver. Simulation results of fractional Chen system verify the effectiveness of proposed scheme.