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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 |
References:
[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. |
[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. |
[3] |
L. J. Cherene, Jr., Set Valued Dynamical Systems and Economic Flow, Lecture Notes in Economics and Mathematical Systems, Springer-Verlag, Berlin/New York, 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), 597-611.
doi: 10.1023/A:1016339216210. |
[5] |
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. |
[6] |
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. |
[7] |
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. |
[8] |
C. Conley, Isolated Invariant Sets and the Morse Index, C.M.B.S. 38, Amer.Math.Soc., Providence, 1978. |
[9] |
M. Hurley, Chain recurrence and attraction in noncompact spaces, Ergodic Theory Dynam. Systems, 11 (1991), 709-729.
doi: 10.1017/S014338570000643X. |
[10] |
M. Hurley, Noncompact chain recurrence and attraction, Proc. Amer. Math. Soc., 115 (1992), 1139-1148.
doi: 10.1090/S0002-9939-1992-1098401-X. |
[11] |
M. Hurley, Chain recurrence, semiflow and gradient, J. Dynam. Differential Equations, 7 (1995), 437-456.
doi: 10.1007/BF02219371. |
[12] |
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. |
[13] |
P. E. Kloeden, Eventual stability in gerneral control systems, J. Differential Equations, 19 (1975), 106-124.
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), 575-586.
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), 35-60.
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), 1-38.
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), 705-724.
doi: 10.1016/S0022-247X(02)00352-9. |
[18] |
Z. Liu, The random case of Conley's theorem, Nonlinearity, 19 (2006), 277-291.
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), 1017-1030.
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), 308-313.
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), 2050-2059.
doi: 10.1016/j.na.2006.06.057. |
[22] |
E. Roxin, Stability in general control systems, J. Differential Equations, 1 (1965), 115-150.
doi: 10.1016/0022-0396(65)90015-X. |
[23] |
K. S. Sibirsky, Introduction to Topological Dynamics, Noordhoff International Publishing, Leyden, The Netherlands, 1975. |
[24] |
J. Tsinias, A Lyapunov description of stability in control systems, Nonlinear Anal., 13 (1989), 63-74.
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), 547-565.
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), 147-154.
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, Berlin/New York, 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), 597-611.
doi: 10.1023/A:1016339216210. |
[5] |
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. |
[6] |
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. |
[7] |
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. |
[8] |
C. Conley, Isolated Invariant Sets and the Morse Index, C.M.B.S. 38, Amer.Math.Soc., Providence, 1978. |
[9] |
M. Hurley, Chain recurrence and attraction in noncompact spaces, Ergodic Theory Dynam. Systems, 11 (1991), 709-729.
doi: 10.1017/S014338570000643X. |
[10] |
M. Hurley, Noncompact chain recurrence and attraction, Proc. Amer. Math. Soc., 115 (1992), 1139-1148.
doi: 10.1090/S0002-9939-1992-1098401-X. |
[11] |
M. Hurley, Chain recurrence, semiflow and gradient, J. Dynam. Differential Equations, 7 (1995), 437-456.
doi: 10.1007/BF02219371. |
[12] |
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. |
[13] |
P. E. Kloeden, Eventual stability in gerneral control systems, J. Differential Equations, 19 (1975), 106-124.
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), 575-586.
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), 35-60.
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), 1-38.
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), 705-724.
doi: 10.1016/S0022-247X(02)00352-9. |
[18] |
Z. Liu, The random case of Conley's theorem, Nonlinearity, 19 (2006), 277-291.
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), 1017-1030.
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), 308-313.
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), 2050-2059.
doi: 10.1016/j.na.2006.06.057. |
[22] |
E. Roxin, Stability in general control systems, J. Differential Equations, 1 (1965), 115-150.
doi: 10.1016/0022-0396(65)90015-X. |
[23] |
K. S. Sibirsky, Introduction to Topological Dynamics, Noordhoff International Publishing, Leyden, The Netherlands, 1975. |
[24] |
J. Tsinias, A Lyapunov description of stability in control systems, Nonlinear Anal., 13 (1989), 63-74.
doi: 10.1016/0362-546X(89)90035-7. |
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