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.

[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.

[3]

L. J. Cherene, Jr., Set Valued Dynamical Systems and Economic Flow, Lecture Notes in Economics and Mathematical Systems,, Springer-Verlag, (1978).

[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.

[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.

[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.

[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.

[8]

C. Conley, Isolated Invariant Sets and the Morse Index,, C.M.B.S. 38, (1978).

[9]

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

[10]

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

[11]

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

[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.

[13]

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

[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.

[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.

[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.

[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.

[18]

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

[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.

[20]

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

[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.

[22]

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

[23]

K. S. Sibirsky, Introduction to Topological Dynamics,, Noordhoff International Publishing, (1975).

[24]

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

show all references

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.

[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.

[3]

L. J. Cherene, Jr., Set Valued Dynamical Systems and Economic Flow, Lecture Notes in Economics and Mathematical Systems,, Springer-Verlag, (1978).

[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.

[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.

[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.

[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.

[8]

C. Conley, Isolated Invariant Sets and the Morse Index,, C.M.B.S. 38, (1978).

[9]

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

[10]

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

[11]

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

[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.

[13]

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

[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.

[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.

[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.

[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.

[18]

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

[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.

[20]

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

[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.

[22]

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

[23]

K. S. Sibirsky, Introduction to Topological Dynamics,, Noordhoff International Publishing, (1975).

[24]

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

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