Research On Algorithm Of Anticontrol And Control Of Chaos In Discrete Dynamical Systems

The problem about control of chaos as well as anticontrol of chaos via feed-back in discrete dynamical systems is studied in this dissertation. Research on control and anticontrol of chaos is a very novel issue. It is important and valuable to chaotify a discrete dynamical system via feed-back control, and so is to control a chaotic discrete system.There are several different goals to chaotify a discrete system via feed-back control. Some problems should be solved as following: to chaotify a discrete system, to make a discrete system hyper-chaotic, to strengthen chaos in discrete system, to weaken or eliminate chaos in discrete system. One way to get these goals is to place the sign and value of Lyapunov exponents in a discrete system. Chen-Lai's algorithm are representative among those very few algorithms to chaotify a discrete system. The main result of Chen-Lai's algorithm and Wang-Chen's algorithm is very important which can place all Lyapunov exponents greater than a given positive constant, c. But it is a regret that this result does not stand just in line with the criterion to judge a discrete system chaotic or not.On the basis of these two algorithms some important problems about how to chaotify a discrete system are put forward and solved in this dissertation. A series of different algorithms to place Lyapunov exponents is proposed. The correspond research as well as some main results consist of several respects as followed:1.An algorithm to chaotify a discrete system: to place only one Lyapunov exponent greater than a given positive constant, c, via some special feed-back while other Lyapunov exponents are none-zero. This result agree with the criterion very well. The algorithm can be used to chaotify a discrete system effectively.2.An algorithm to make a discrete system hyperchaotic: to place only two Lyapunov exponent greater than a given positive constant, c, via some special feed-back while other Lyapunov exponents are none-zero. The problems about how to make a discrete system chaotic or hyperchaotic are united to one: how to place some Lyapunov exponents greater than a given positive constant, c. The algorithm even can be evolved to Chen-Lai's algorithm and Wang-Chen's algorithm.3. An algorithm to place only those greatest Lyapunov exponents of discrete systems greater than a given positive constant, c: It is discussed that smaller feed-back gain than that in Chen-Lai's algorithm and Wang-Chen's algorithm can place Lyapunov exponents positive. The problem about placing only one or two greatest Lyapunov exponents is solved and these results agree with the criterion very well. On the basis of these results, Chen-Lai's algorithm and Wang-Chen's algorithm are improved on three different level.4. An algorithm to strengthen or weaken chaos in discrete systems: the relation between Lyapunov exponents of discrete systems and elements of Jacobian matrix is studied. Some new conditions on elements of Jacobian matrix and eigenvalues is proposed to estimate the signs and values of Lyapunov exponents. Some new none-diagonal feed-back are introduced to arrange the elements of Jacobian matrix. The Lyapunov exponents are placed to be different range of value via special feed-back, and as a result chaos in discrete systems is strengthened or weakened.5. An algorithm to place all Lyapunov exponents of discrete systems precisely. The problem about how to place all Lyapunov exponents of discrete systems precisely is discussed and solved.Correspond proofs and simulation experiements all algorithms proposed in this dissertation was presented. The simulation results show the effectiveness of these algorithms.