Advanced Search
Article Contents
Article Contents

Conley's theorem for dispersive systems

Abstract Related Papers Cited by
  • In this article, we study Conley's theorem about the chain recurrence in dynamical systems, that is, the chain recurrent set of continuous map $f$ is the complement of union of $B_{U}(A)-A$, where $A$ is an attractor and $B_{U}(A)$ is a basin of $A$. In this paper, we generalize this theorem to dispersive systems on noncompact spaces.
    Mathematics Subject Classification: Primary: 37B20; Secondary: 37B25, 37B35.


    \begin{equation} \\ \end{equation}
  • [1]

    A. Bacciotti and N. Kalouptsidis, Topological dynamics of control systems: Stability and attraction, Nonlinear Anal., 10 (1986), 547-565.doi: 10.1016/0362-546X(86)90142-2.


    U. Bronstein and A. Ya. Kopanskii, Chain recurrence in dynamical systems without uniqueness, Nonlinear Anal., 12 (1988), 147-154.doi: 10.1016/0362-546X(88)90031-4.


    L. J. Cherene, Jr., Set Valued Dynamical Systems and Economic Flow, Lecture Notes in Economics and Mathematical Systems, Springer-Verlag, Berlin/New York, 1978.


    S. Choi, C. Chu and J.-S. Park, Chain recurrent sets for flows on non-compact spaces, J. Dynam. Differential Equations, 12 (2002), 597-611.doi: 10.1023/A:1016339216210.


    H.-Y. Chu, Chain recurrence for multi-valued dynamical systems on noncompact spaces, Nonlinear Anal., 61 (2005), 715-723.doi: 10.1016/j.na.2005.01.024.


    H.-Y. Chu, Strong centers of attraction for multi-valued dynamical systems on noncompact spaces, Nonlinear Anal., 68 (2008), 2479-2486.doi: 10.1016/j.na.2007.01.072.


    H.-Y. Chu and J.-S. Park, Attractors for relations in $\sigma$-compact spaces, Topology Appl., 148 (2005), 201-212.doi: 10.1016/j.topol.2003.05.009.


    C. Conley, Isolated Invariant Sets and the Morse Index, C.M.B.S. 38, Amer.Math.Soc., Providence, 1978.


    M. Hurley, Chain recurrence and attraction in noncompact spaces, Ergodic Theory Dynam. Systems, 11 (1991), 709-729.doi: 10.1017/S014338570000643X.


    M. Hurley, Noncompact chain recurrence and attraction, Proc. Amer. Math. Soc., 115 (1992), 1139-1148.doi: 10.1090/S0002-9939-1992-1098401-X.


    M. Hurley, Chain recurrence, semiflow and gradient, J. Dynam. Differential Equations, 7 (1995), 437-456.doi: 10.1007/BF02219371.


    P. E. Kloeden, Asymptotic invariance and limit sets of general control systems, J. Differential Equations, 19 (1975), 91-105.doi: 10.1016/0022-0396(75)90021-2.


    P. E. Kloeden, Eventual stability in gerneral control systems, J. Differential Equations, 19 (1975), 106-124.doi: 10.1016/0022-0396(75)90022-4.


    K. B. Lee and J.-S. Park, Chain recurrence and attractions in general dynamical systems, Commun. Korean Math. Soc., 22 (2007), 575-586.doi: 10.4134/CKMS.2007.22.4.575.


    D. Li, Morse decompositions for general dynamical systems and differential inclusions with applications to control systems, SIAM J. Control Optim., 46 (2007), 35-60.doi: 10.1137/060662101.


    D. Li and P. E. Kloeden, On the dynamics of nonautonomous periodic general dynamical systems and differential inclusions, J. Differential Equations, 224 (2006), 1-38.doi: 10.1016/j.jde.2005.07.012.


    D. Li and X. Zhang, On dynamical properties of general dynamical systems and differential inclusions, J. Math. Anal. Appl., 274 (2002), 705-724.doi: 10.1016/S0022-247X(02)00352-9.


    Z. Liu, The random case of Conley's theorem, Nonlinearity, 19 (2006), 277-291.doi: 10.1088/0951-7715/19/2/002.


    Z. Liu, The random case of Conley's theorem : II. The complete Lyapunov function, Nonlinearity, 20 (2007), 1017-1030.doi: 10.1088/0951-7715/20/4/012.


    J. W. Nieuwenhuis, Some remarks on set-valued dynamical systems, J. Aust. Math. Soc., 22 (1981), 308-313.doi: 10.1017/S0334270000002654.


    J.-S. Park, D. S. Kang and H.-Y. Chu, Stabilities in multi-valued dynamical systems, Nonlinear Anal., 67 (2007), 2050-2059.doi: 10.1016/j.na.2006.06.057.


    E. Roxin, Stability in general control systems, J. Differential Equations, 1 (1965), 115-150.doi: 10.1016/0022-0396(65)90015-X.


    K. S. Sibirsky, Introduction to Topological Dynamics, Noordhoff International Publishing, Leyden, The Netherlands, 1975.


    J. Tsinias, A Lyapunov description of stability in control systems, Nonlinear Anal., 13 (1989), 63-74.doi: 10.1016/0362-546X(89)90035-7.

  • 加载中

Article Metrics

HTML views() PDF downloads(83) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint