Analysis And Control For Several New Continuous Chaos Systems

Based on Lyapunov theory, an new auxiliary search method is put forward for aspecial kind of Lorenz-type autonomous differential equations. The basic idea is toobtain the expected form of system equations, by means of constructing the derivativeof Lyapunov function which satisfies some determinate conditions. The new methodis very simple and convenient. The searching range is extremely reduced for findingchaotic behavior in complex mathematical model. Utilizing the new method, three newLorenz-type chaotical system is found. They are three dimensional quadratic dissipa-tive system with three or four quadratic terms and negative main diagonal elements ofcoefficient matrix in the linear part of the system. Therefore, they are similar to theexisting Lorenz-type systems, such as Lorenz system, Ro¨ssler system, Chen system, lu¨system and LC system, but explicitly different from them. The dynamical behaviors areinvestigated in terms of local and global stability analysis, phase trajectories, lyapunovexponents, Poincare′mapping and bifurcation diagrams. The results show there is co-existence phenomena of stable equilibrium, limit loop and chaotic attractor in the firstsystem, a 3-scroll chaotical attractor in the second, and a multilayer taper attractor in thethird. However, there are more complex chaotic attractors in the new systems. Further-more, the global attractive sets are found for the first and the second system, and strictmathematical proofs are given.Stabilization for unstable equilibria of the new systems and LC system is studied.Globally exponential stabilization of euilibria are discussed based on linear or nonlin-ear state feedback, and a series of simple algebraic criteria are obtained. The adaptivecontrol technique of the systems with unknown parameters is considered. Furthermore,by means of Lyapunov stability theory, an asymptotically stable criterion for a generalchaotical system is obtained. Finally, approach of controller design for chaos controlvia impulsive control technique is discussed, and the relation of impulse interval andcontroller gain is given if the controlled system is stable.The chaos synchronization of the new systems and LC system is investigated. Non-linear feedback controllers are presented to achieve globally exponential synchronization for two same chaotic systems with different initial conditions. Several algebraic suffi-cient conditions are developed. Adaptive synchronization strategy is introduced whenthe parameters of systems are all unknown. An adaptive controller is designed, whichcan guarantee globally asymptotic synchronization. If the unknown parameters varywith time, assuming that the varying parameters are bounded but the boundedness is un-known, the adaptive sliding mode type controller is designed, and globally asymptoticrobust synchronization is also realized.Dynamical behaviors of several time-delayed chaotical system is studied. Basedon transcendant characteristic equation associated with the Halanay inequality, the localstability criteria, delay independent or delay dependent, is obtained for the equilibria ofdelay logistic equation. The stability of bifurcation periodic solutions and the directionof Hopf bifurcation are determined by applying the normal form theory and the centermanifold theorem. Chaotic behavior of the parameterized Logistic differential systemswith a single delay is detected by numerical examples. A new chaotical time-delayrecurrent neural network with four neurons is put forward. Its dynamical propertyis investigated via both Lyapunov theory and numerical simulation method. Finally,a simple neutral chaotical system is presented. To our best knowledge, no neutralchaotical system is reported by far.