Multi-wing Multi-scroll Chaotic Systems Generation,Synchronization And Its Application To Secure Communications

Nonlinear dynamics, commonly called the chaos theory, changes the scientific way of looking at the dynamics of natural and social systems, which has been intensively studied over the past several decades. Chaos is a kind of characteristics of nonlinear systems, which is a bounded unstable dynamic behavior that exhibits sensitive dependence on initial conditions and includes infinite unstable periodic motions. Although it appears to be stochastic, it occurs in a deterministic nonlinear system under deterministic conditions.Generating a chaotic system with a more complicated topological structure such as multi-wing or multi-scroll attractors becomes a desirable task and sometimes a key issue for many engineering applications. In this endeavor, there are two major efforts as follows:based on generalizing smooth and nonsmooth nonlinear function autonomous systems with multi-wing attractor and generalizing nonsmooth nonlinear function autonomous systems with multi-scroll attractors. Meanwhile, based on linear sciences, chaos was often regarded as a harmful and undesirable characteristic for most physics and engineering applications due to its unpredictable nature. However, in the last two decades, and since the pioneer work of chaos control, and chaos synchronization, chaos proved to be of effective use for secure communications. Based on the complex chaos system modeling and synchronization theory, the main innovative works of this dissertation are as follows:(1) In this dissertation we introduce a three-dimensional (3D) exponential-type chaotic system and a3D hyperbolic-type chaotic system with only five terms including one nonlinear term in the form of exponential function and hyperbolic function respectively. The two systems can generate a two-wing chaotic attractor when all of equilibria are stable. Compared with other3D chaotic systems, not only the terms of the two systems are less, but also the range of chaos is wider when the parameter varies. Based on the same method, this dissertation proposes two-wing exponential-type hyperchaotic system and hyperbolic-type hyperchaotic system equipped with a nonlinear term in the form of exponential function and hyperbolic function respectively. The two systems are autonomous with a unique equilibrium, especially, a Hopf bifurcation occurs at this equilibrium. (2) In this dissertation, we propose a group of four-dimensional (4D) autonomous chaotic systems with five real equilibria. These systems can generate one-, two-, three-and four-wing attractors with variation of a single parameter, and the multi-wing type of the attractors can be displayed in all directions. In this dissertation we initiate a new approach for oblique grid multi-scroll hyperchaos generation. Through introducing two piecewise-linear triangular wave functions into a three-dimensional spiral chaotic Colpitts oscillator model, a4D grid multi-scroll hyperchaotic system is constructed. By adjusting a build-in parameter in a variable of one triangle wave function, the control of the gradient of the multi-scroll grid is achieved. Meanwhile, by deploying the zero points of the two triangular wave functions to extend the saddle-focus equilibrium points with index-2in phase space, the scroll numbers not only increase along with the number of turning points, but also can generate arbitrary multiples of products.(3) Projective synchronization (PS) of two chaotic systems with fully uncertain parameters is investigated in this dissertation. Based on Lyapunov stability theory and Barbalat’s lemma, a new adaptive controller with parameter update laws is designed to complete synchronization (CS) and antiphase synchronization (AS) between two chaotic systems asymptotically and globally, including two identical hyperbolic-type chaotic systems and two different chaotic systems. Based on PS technique, in this dissertation we also propose a new synchronization, complete switched modified function projective synchronization (CSMFPS), for two different chaotic systems, where the drive and response systems could be complete switched synchronized to a function matrix. The unpredictability of the function matrix in CSMFPS can additionally strengthen the security of communications. Finally, the five-term exponential-type chaotic system is taken for example and the corresponding numerical simulations are presented to verify the effectiveness and feasibility of the proposed control scheme. Most of synchronization methods mainly concerned the synchronization that needed to construct the appropriate Lyapunov functions. As is well known, the construction of the appropriate Lyapunov functions is still a difficult issue. An extended control scheme is presented by using Jacobin matrix method, which realizes synchronization of two different4D chaotic systems, and the nonlinear terms in the drive system are not removed. To illustrate the effectiveness of the proposed scheme, a numerical example based on the exponential-type hyperchaotic system is presented. (4) A modified chaotic masking secure communication scheme applying complete switched projective synchronization of the exponential-type hyperchaotic system is proposed in this dissertation. In the chaotic systems of the sender and the receiver, the feedback of the mixed signal of the useful signal and chaotic signal is introduced, respectively. It can improve the precision of the synchronization, so as to overcome some shortcomings of the traditional chaos masking and realize the encryption and recovery of the useful signal. Secondly, another secure communication scheme based on a four-wing chaotic system with external disturbance via a convenient robust high-order sliding mode adaptative controller is also discussed in this dissertation. By parameter modulation theory and Lyapunov stability theory, synchronization and secure communication between transmitter and receiver is achieved and two message signals are recovered. In addition, the gains of the receiver system can be adjusted continually, the unknown parameters can be identified precisely and the external disturbance can be suppressed simultaneously by the proposed adaptative controller.