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Park, Jong-Suh; Kang, Dong Seung; Chu, Hahng-Yun

Stabilities in multi-valued dynamical systems. (English) Zbl 1121.37019

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 67, No. 7, 2050-2059 (2007).
Summary: For the study of the various stabilities in multivalued dynamical systems, we obtain some necessary and sufficient conditions of the notions of stability and Lyapunov stability, and investigate the connection between the concepts of attractions and the suitable versions of limit sets. Also we consider the notion of characteristic \(0^{+}\) in multivalued dynamical systems and obtain necessary and sufficient conditions for the concept of characteristic \(0^{+}\).

MSC:

37B25 Stability of topological dynamical systems
37B35 Gradient-like behavior; isolated (locally maximal) invariant sets; attractors, repellers for topological dynamical systems

Keywords:

stability; Lyapunov stability; characteristic \(0^{+}\); disperse systems; attractors; limit sets
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References:

[1] Ahmad, K.; Hamoui, A., Dynamical systems of proper characteristic 0, Dynam. Systems Appl., 6, 11-20 (1997) · Zbl 0887.58027
[2] Ahmad, S., Dynamical systems of characteristic \(0^+\), Pacific J. Math., 32, 561-574 (1970) · Zbl 0176.20402
[3] Akin, E., (The General Topology of Dynamical Systems. The General Topology of Dynamical Systems, Graduate Studies in Mathematics, vol. 71 (1993), Amer. Math. Soc.: Amer. Math. Soc. Providence) · Zbl 0781.54025
[4] Akin, E.; Miller, W., Invariant relations for set-valued dynamical systems, Trans. Amer. Math. Soc. 351, 2, 1203-1225 (1999) · Zbl 0928.54039
[5] Bronstein, U.; Kopanskii, A. Ya., Chain recurrence in dynamical systems without uniqueness, Nonlinear Anal. TMA, 12, 147-154 (1988) · Zbl 0644.58008
[6] Bacciotti, A.; Kalouptsidis, N., Topological dynamics of control systems: Stability and attraction, Nonlinear Anal. TMA, 10, 547-565 (1986) · Zbl 0611.93052
[7] Bhatia, N. P.; Szegö, G. P., Stability Theory of Dynamical Systems (1970), Springer-Verlag: Springer-Verlag New York · Zbl 0213.10904
[8] Bianchini, R. M.; Zecca, P., On the attainable set at time t for multivalued differential equations, Nonlinear Anal. TMA, 5, 1053-1059 (1981) · Zbl 0469.49028
[9] Cherene, L. J., (Set Valued Dynamical Systems and Economic Flow. Set Valued Dynamical Systems and Economic Flow, Lecture Notes in Economics and Mathematical Systems, vol. 158 (1978), Springer-Verlag: Springer-Verlag Berlin, New York) · Zbl 0396.90003
[10] Chu, H. Y., Chain recurrence for multi-valued dynamical systems on noncompact spaces, Nonlinear Anal. TMA, 61, 715-723 (2005) · Zbl 1064.37016
[11] Chu, H. Y.; Park, J. S., Attractors for relations in \(\sigma \)-compact spaces, Topology Appl., 148, 201-212 (2005) · Zbl 1073.37014
[12] Kloeden, P. E., Asymptotic invariance and limit sets of general control systems, J. Differential Equations, 19, 91-105 (1975) · Zbl 0306.93024
[13] Kloeden, P. E., Eventual stability in general control systems, J. Differential Equations, 19, 106-124 (1975) · Zbl 0294.93007
[14] Li, D., On dynamical stability in general dynamical systems, J. Math. Anal. Appl., 263, 455-478 (2001) · Zbl 1109.37300
[15] Li, D.; Zhang, X., On dynamical properties of general dynamical systems and differential inclusions, J. Math. Anal. Appl., 274, 705-724 (2002) · Zbl 1007.37005
[16] McGehee, R., Attractors for closed relations on compact Hausdorff space, Indiana Univ. Math. J., 41, 1165-1209 (1992) · Zbl 0767.58032
[17] Nieuwenhuis, J. W., Some remarks on set-valued dynamical systems, J. Aust. Math. Soc. Ser. B, 22, 308-313 (1981) · Zbl 0464.49028
[18] Roxin, E., Stability in general control systems, J. Differential Equations, 1, 115-150 (1965) · Zbl 0143.32302
[19] Sibirsky, K. S., Introduction to Topological Dynamics (1975), Noordhoff International Publishing: Noordhoff International Publishing Leyden, The Netherlands · Zbl 0297.54001
[20] Szegö, G. P.; Treccani, G., Flow without uniqueness near a compact strongly invariant set, Boll. Unione. Mat. Ital., 2, 113-124 (1969) · Zbl 0184.26902
[21] Tsinias, J., A Lyapunov description of stability in control systems, Nonlinear Anal. TMA, 13, 63-74 (1989) · Zbl 0695.93083
[22] Tsinias, J.; Kalouptsidis, N., Prolongations and stability analysis via Lyapunov functions of dynamical polysystems, Math. Syst. Theory, 20, 215-233 (1987) · Zbl 0642.93052
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