Research On Some Issues Of Anti-control In Dynamical Systems

Since the1960s, as an important branch of dynamical system, chaos has captured the attention of many researchers. In recent years, with in-depth study, scientists gradually realized the possibility and importance on the basis of modern control theory to solve the primary problem of chaos, among which the anti-control of chaos is one of the effective ways to solve problems by combining modern control theory and chaos theory.Anti-Control is usually divided into anti-control of continuous-time systems and anti-control of discrete-time systems. For anti-control of continuous-time systems so far has not been satisfactorily resolved. In1979, Rossler firstly put forward the concept of hyperchaos. Hyperchaos refers to a chaotic system with two or more than two positive Lyapunov exponents. In recent years, a large number of results show that the phase trajectory of hyperchaotic system can be stretched and folded movement in more directions, and its dynamic behavior is more complex than the general chaotic system. Therefore, hyperchaotic system has a broader application prospect. However, according to the existing literatures, one can design the hyperchaotic system with maximum three positive Lyapunov exponents up to now. This is because, with the increasing of dimension, it will be more and more difficult to construct higher dimensional hyperchaotic systems with four or more positive Lyapunov exponents by using the traditional trial-and-error approach without any theoretical guidelines. Therefore, how to establish a general approach to design a hyperchaotic system with a desired number of positive Lyapunov exponents has been a long-standing open technical problem over the last three decades.The first part of the thesis is mainly aimed at some hot issues of the anti-control of hyperchaotic system. The specific contents are as follows:(1) A systematic methodology for constructing continuous-time autonomous hyperchaotic systems with multiple positive Lyapunov exponents is proposed. To overcome the essential difficulty in balancing between local instability and global convergence, a new methodology for designing a dissipative hyperchaotic system with a desired number of positive Lyapunov exponents is investigated. A general design principle and the corresponding implementation steps are developed. Four representative examples are shown to validate the proposed principle and implementation scheme. Moreover, a hyperchaotic circuit is constructed to verify a6-dimensional hyperchaotic system with four positive Lyapunov exponents. Comparing with the traditional trial-and-error approach, the proposed method can design various dissipative hyperchaotic systems with any desired number of positive Lyapunov exponents in a systematic way.(2) Based on the anti-control principle, a unified chaotification framework for generating desired higher-dimensional dissipative and conservative hyperchaotic system by using a single-parameter controller is proposed. In particular, a block-diagonal matrix is introduced to design the nominal system matrix to guarantee the effective control goal via similarity transformation. In detail, the proposed approach assigns the closed-loop poles of controlled system to allocate the corresponding numbers of eigenvalues with positive real parts of two classes of saddle-focus equilibriums to be n-1and n-2, respectively. For the no degeneration case, the number of positive Lyapunov exponent in the controlled system is given byL=min{n-1,n-2}=n-2, which is the largest number of Lyapunov exponents of n-dimensional hyperchaotic system. Several representative examples are also given to validate the proposed method. Finally, the Digital Signal Processor (DSP) is employed to implement the above10-dimensional dissipative hyperchaotic systems and the experimental observation is also given. This method provides a new approach to constructing high dimensional dissipative and conservative autonomous hyperchaotic system.(3) A new and unified approach for designing desirable dissipative hyperchaotic systems is introduced. Based on the anti-control principle of continuous-time systems, a nominal system of n(n≥5) independent first-order linear differential equations are coupled through all state variables, making the controlled system be in a closed-loop cascade-coupling form, where each equation contains only two state variables therefore the system is quite simple. Based on this setting, a simple model for dissipative hyperchaotic systems is constructed, with an adjustable parameter which can ensure the dissipation of the system. In the closed-loop cascade-coupling form, it is shown that all the eigenvalues are symmetrically distributed in a circumferential manner. Consequently, a universal law is derived on the relationship of the number of positive Lyapunov exponents and the number of positive real parts of its Jacobian eigenvalues. For the above-mentioned simple model, the number of positive Lyapunov exponents for any n-dimensional dissipative hyperchaotic system is given by N=round((n-1)/2),n≥5. Therefore, in theory, the system can generate any desired number of positive Lyapunov exponents as long as the dimension of the system is sufficiently high. Thus, the proposed method provides a new approach for purposefully constructing desirable dissipative hyperchaotic systems. Finally, two examples are given to demonstrate the feasibility of the proposed design method.(4) A new four-dimensional hyperchaotic system is built by adding a linear state feedback controller into a smooth quadratic three-order chaotic system. Some of its basic dynamical properties are studied in detail. Such as the feature of equilibrium point, the hyperchaotic attractor, the Lyapunov exponent spectrum and the bifurcation diagram. Numerical simulations show that the new system’s behavior can be periodic, complex periodic, chaotic and hyperchaotic as the parameter varies. Finally, an electronic circuit is designed to realize the chaotic system. Experimental observations show very good agreement with the simulation results.The second part of this dissertation is mainly for circuit implementation of the discrete-time chaotic systems. In the existing literature, there is no detailed report about the circuit design and experimental implementation for the anti-control of discrete-time systems with sine and triangular wave modulus functions. Therefore, it is also an important issue in the research of this dissertation. To be specific, two aspects are investigated as follows:(1) Based on the Chen-Lai algorithm, two types of anti-control of discrete-time systems both with triangular wave and sine wave functions on the basis of a finite region are introduced, respectively. Then, theoretical proofs are further provided in the sense of Li-Yorke. (2) Furthermore, taken three-dimensional discrete-time systems as typical examples, the corresponding simulation results are presented. Finally, Acorrding to the the sufficient conditions from the theorem and the dynamic range of the physical devices, the anti-control circuits of discrete-time systems are designed. The experiment results are in agreement with the results of simulation, which confirms the feasibility of the theorem.This work was supported by the National Natural Science Foundation of China under Grants61172023, the Specialized Research Foundation of Doctoral Subjects of Education Ministry under Grant20114420110003.