Conley's theorem for dispersive systems

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

Conley's theorem for dispersive systems

Hahng-Yun Chu ^1, , Se-Hyun Ku ^1, and Jong-Suh Park ^1,

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

Abstract
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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.

Keywords: Chain recurrence, basin of attractor, compact-valued dispersive systems., attractor, dispersive systems.

Mathematics Subject Classification: Primary: 37B20; Secondary: 37B25, 37B3.

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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