# American Institute of Mathematical Sciences

April  2015, 8(2): 313-321. doi: 10.3934/dcdss.2015.8.313

## Conley's theorem for dispersive systems

 1 Department of Mathematics, Chungnam National University, 79, Daehak-ro, Yuseong-gu, Daejeon 305-764, South Korea, South Korea, South Korea

Received  April 2013 Revised  November 2013 Published  July 2014

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.
Citation: Hahng-Yun Chu, Se-Hyun Ku, Jong-Suh Park. Conley's theorem for dispersive systems. Discrete & Continuous Dynamical Systems - S, 2015, 8 (2) : 313-321. doi: 10.3934/dcdss.2015.8.313
##### References:
 [1] A. Bacciotti and N. Kalouptsidis, Topological dynamics of control systems: Stability and attraction,, Nonlinear Anal., 10 (1986), 547. doi: 10.1016/0362-546X(86)90142-2. Google Scholar [2] U. Bronstein and A. Ya. Kopanskii, Chain recurrence in dynamical systems without uniqueness,, Nonlinear Anal., 12 (1988), 147. doi: 10.1016/0362-546X(88)90031-4. Google Scholar [3] L. J. Cherene, Jr., Set Valued Dynamical Systems and Economic Flow, Lecture Notes in Economics and Mathematical Systems,, Springer-Verlag, (1978). Google Scholar [4] S. Choi, C. Chu and J.-S. Park, Chain recurrent sets for flows on non-compact spaces,, J. Dynam. Differential Equations, 12 (2002), 597. doi: 10.1023/A:1016339216210. Google Scholar [5] H.-Y. Chu, Chain recurrence for multi-valued dynamical systems on noncompact spaces,, Nonlinear Anal., 61 (2005), 715. doi: 10.1016/j.na.2005.01.024. Google Scholar [6] H.-Y. Chu, Strong centers of attraction for multi-valued dynamical systems on noncompact spaces,, Nonlinear Anal., 68 (2008), 2479. doi: 10.1016/j.na.2007.01.072. Google Scholar [7] H.-Y. Chu and J.-S. Park, Attractors for relations in $\sigma$-compact spaces,, Topology Appl., 148 (2005), 201. doi: 10.1016/j.topol.2003.05.009. Google Scholar [8] C. Conley, Isolated Invariant Sets and the Morse Index,, C.M.B.S. 38, (1978). Google Scholar [9] M. Hurley, Chain recurrence and attraction in noncompact spaces,, Ergodic Theory Dynam. Systems, 11 (1991), 709. doi: 10.1017/S014338570000643X. Google Scholar [10] M. Hurley, Noncompact chain recurrence and attraction,, Proc. Amer. Math. Soc., 115 (1992), 1139. doi: 10.1090/S0002-9939-1992-1098401-X. Google Scholar [11] M. Hurley, Chain recurrence, semiflow and gradient,, J. Dynam. Differential Equations, 7 (1995), 437. doi: 10.1007/BF02219371. Google Scholar [12] P. E. Kloeden, Asymptotic invariance and limit sets of general control systems,, J. Differential Equations, 19 (1975), 91. doi: 10.1016/0022-0396(75)90021-2. Google Scholar [13] P. E. Kloeden, Eventual stability in gerneral control systems,, J. Differential Equations, 19 (1975), 106. doi: 10.1016/0022-0396(75)90022-4. Google Scholar [14] K. B. Lee and J.-S. Park, Chain recurrence and attractions in general dynamical systems,, Commun. Korean Math. Soc., 22 (2007), 575. doi: 10.4134/CKMS.2007.22.4.575. Google Scholar [15] D. Li, Morse decompositions for general dynamical systems and differential inclusions with applications to control systems,, SIAM J. Control Optim., 46 (2007), 35. doi: 10.1137/060662101. Google Scholar [16] D. Li and P. E. Kloeden, On the dynamics of nonautonomous periodic general dynamical systems and differential inclusions,, J. Differential Equations, 224 (2006), 1. doi: 10.1016/j.jde.2005.07.012. Google Scholar [17] D. Li and X. Zhang, On dynamical properties of general dynamical systems and differential inclusions,, J. Math. Anal. Appl., 274 (2002), 705. doi: 10.1016/S0022-247X(02)00352-9. Google Scholar [18] Z. Liu, The random case of Conley's theorem,, Nonlinearity, 19 (2006), 277. doi: 10.1088/0951-7715/19/2/002. Google Scholar [19] Z. Liu, The random case of Conley's theorem : II. The complete Lyapunov function,, Nonlinearity, 20 (2007), 1017. doi: 10.1088/0951-7715/20/4/012. Google Scholar [20] J. W. Nieuwenhuis, Some remarks on set-valued dynamical systems,, J. Aust. Math. Soc., 22 (1981), 308. doi: 10.1017/S0334270000002654. Google Scholar [21] J.-S. Park, D. S. Kang and H.-Y. Chu, Stabilities in multi-valued dynamical systems,, Nonlinear Anal., 67 (2007), 2050. doi: 10.1016/j.na.2006.06.057. Google Scholar [22] E. Roxin, Stability in general control systems,, J. Differential Equations, 1 (1965), 115. doi: 10.1016/0022-0396(65)90015-X. Google Scholar [23] K. S. Sibirsky, Introduction to Topological Dynamics,, Noordhoff International Publishing, (1975). Google Scholar [24] J. Tsinias, A Lyapunov description of stability in control systems,, Nonlinear Anal., 13 (1989), 63. doi: 10.1016/0362-546X(89)90035-7. Google Scholar

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##### References:
 [1] A. Bacciotti and N. Kalouptsidis, Topological dynamics of control systems: Stability and attraction,, Nonlinear Anal., 10 (1986), 547. doi: 10.1016/0362-546X(86)90142-2. Google Scholar [2] U. Bronstein and A. Ya. Kopanskii, Chain recurrence in dynamical systems without uniqueness,, Nonlinear Anal., 12 (1988), 147. doi: 10.1016/0362-546X(88)90031-4. Google Scholar [3] L. J. Cherene, Jr., Set Valued Dynamical Systems and Economic Flow, Lecture Notes in Economics and Mathematical Systems,, Springer-Verlag, (1978). Google Scholar [4] S. Choi, C. Chu and J.-S. Park, Chain recurrent sets for flows on non-compact spaces,, J. Dynam. Differential Equations, 12 (2002), 597. doi: 10.1023/A:1016339216210. Google Scholar [5] H.-Y. Chu, Chain recurrence for multi-valued dynamical systems on noncompact spaces,, Nonlinear Anal., 61 (2005), 715. doi: 10.1016/j.na.2005.01.024. Google Scholar [6] H.-Y. Chu, Strong centers of attraction for multi-valued dynamical systems on noncompact spaces,, Nonlinear Anal., 68 (2008), 2479. doi: 10.1016/j.na.2007.01.072. Google Scholar [7] H.-Y. Chu and J.-S. Park, Attractors for relations in $\sigma$-compact spaces,, Topology Appl., 148 (2005), 201. doi: 10.1016/j.topol.2003.05.009. Google Scholar [8] C. Conley, Isolated Invariant Sets and the Morse Index,, C.M.B.S. 38, (1978). Google Scholar [9] M. Hurley, Chain recurrence and attraction in noncompact spaces,, Ergodic Theory Dynam. Systems, 11 (1991), 709. doi: 10.1017/S014338570000643X. Google Scholar [10] M. Hurley, Noncompact chain recurrence and attraction,, Proc. Amer. Math. Soc., 115 (1992), 1139. doi: 10.1090/S0002-9939-1992-1098401-X. Google Scholar [11] M. Hurley, Chain recurrence, semiflow and gradient,, J. Dynam. Differential Equations, 7 (1995), 437. doi: 10.1007/BF02219371. Google Scholar [12] P. E. Kloeden, Asymptotic invariance and limit sets of general control systems,, J. Differential Equations, 19 (1975), 91. doi: 10.1016/0022-0396(75)90021-2. Google Scholar [13] P. E. Kloeden, Eventual stability in gerneral control systems,, J. Differential Equations, 19 (1975), 106. doi: 10.1016/0022-0396(75)90022-4. Google Scholar [14] K. B. Lee and J.-S. Park, Chain recurrence and attractions in general dynamical systems,, Commun. Korean Math. Soc., 22 (2007), 575. doi: 10.4134/CKMS.2007.22.4.575. Google Scholar [15] D. Li, Morse decompositions for general dynamical systems and differential inclusions with applications to control systems,, SIAM J. Control Optim., 46 (2007), 35. doi: 10.1137/060662101. Google Scholar [16] D. Li and P. E. Kloeden, On the dynamics of nonautonomous periodic general dynamical systems and differential inclusions,, J. Differential Equations, 224 (2006), 1. doi: 10.1016/j.jde.2005.07.012. Google Scholar [17] D. Li and X. Zhang, On dynamical properties of general dynamical systems and differential inclusions,, J. Math. Anal. Appl., 274 (2002), 705. doi: 10.1016/S0022-247X(02)00352-9. Google Scholar [18] Z. Liu, The random case of Conley's theorem,, Nonlinearity, 19 (2006), 277. doi: 10.1088/0951-7715/19/2/002. Google Scholar [19] Z. Liu, The random case of Conley's theorem : II. The complete Lyapunov function,, Nonlinearity, 20 (2007), 1017. doi: 10.1088/0951-7715/20/4/012. Google Scholar [20] J. W. Nieuwenhuis, Some remarks on set-valued dynamical systems,, J. Aust. Math. Soc., 22 (1981), 308. doi: 10.1017/S0334270000002654. Google Scholar [21] J.-S. Park, D. S. Kang and H.-Y. Chu, Stabilities in multi-valued dynamical systems,, Nonlinear Anal., 67 (2007), 2050. doi: 10.1016/j.na.2006.06.057. Google Scholar [22] E. Roxin, Stability in general control systems,, J. Differential Equations, 1 (1965), 115. doi: 10.1016/0022-0396(65)90015-X. Google Scholar [23] K. S. Sibirsky, Introduction to Topological Dynamics,, Noordhoff International Publishing, (1975). Google Scholar [24] J. Tsinias, A Lyapunov description of stability in control systems,, Nonlinear Anal., 13 (1989), 63. doi: 10.1016/0362-546X(89)90035-7. Google Scholar
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