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Flows of characteristic \(0^+\) in impulsive semidynamical systems. (English) Zbl 1112.37014

Summary: In this paper, as in the paper of E. M. Bonotto and M. Federson [J. Math. Anal. Appl. 326, No. 2, 869–881 (2006; Zbl 1162.37008)], we continue to study the dynamics of flows defined in impulsive semidynamical systems \((X,\pi;M,I)\), where \(X\) is a metric space, \((X,\pi)\) is a semidynamical system, \(M\) denotes an impulsive set and \(I\) is an impulsive perator. We generalize some results of nonimpulsive flows of characteristic \(0^+(Ch0^+)\) for systems with impulses. In particular, we state conditions so that the limit set of an impulsive system of \(Ch0^+\) is either a periodic orbit or a single rest point. We also give conditions for a subset \(H\) in \((X,\pi;M,I)\) to be globally asymptotically stable in the impulsive system, provided the flow is of \(Ch0^+\).

MSC:

37B99 Topological dynamics
54H20 Topological dynamics (MSC2010)

Citations:

Zbl 1162.37008
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References:

[1] Ahmad, S., Dynamical systems of characteristic \(0^+\), Pacific J. Math., 32, 561-574 (1970) · Zbl 0176.20402
[2] E.M. Bonotto, M. Federson, Topological conjugation and asymptotic stability in impulsive semidynamical systems, J. Math. Anal. Appl. (2006), doi:10.1016/j.jmaa.2006.03.042; E.M. Bonotto, M. Federson, Topological conjugation and asymptotic stability in impulsive semidynamical systems, J. Math. Anal. Appl. (2006), doi:10.1016/j.jmaa.2006.03.042 · Zbl 1162.37008
[3] Ciesielski, K., Sections in semidynamical systems, Bull. Pol. Acad. Sci. Math., 40, 297-307 (1992) · Zbl 0801.54034
[4] Ciesielski, K., The Poincaré-Bendixson theorem for two-dimensional semiflows, Topol. Methods Nonlinear Anal., 3, 163-178 (1994) · Zbl 0839.54028
[5] Ciesielski, K., On semicontinuity in impulsive dynamical systems, Bull. Pol. Acad. Sci. Math., 52, 71-80 (2004) · Zbl 1098.37016
[6] Ciesielski, K., On stability in impulsive dynamical systems, Bull. Pol. Acad. Sci. Math., 52, 81-91 (2004) · Zbl 1098.37017
[7] Kaul, S. K., On impulsive semidynamical systems, J. Math. Anal. Appl., 150, 1, 120-128 (1990) · Zbl 0711.34015
[8] Kaul, S. K., On impulsive semidynamical systems II. Recursive properties, Nonlinear Anal., 16, 635-645 (1991) · Zbl 0724.34010
[9] Kaul, S. K., On impulsive semidynamical systems III: Lyapunov stability, (Recent Trends in Differential Equations. Recent Trends in Differential Equations, Ser. Appl. Anal., vol. 1 (1992), World Scientific: World Scientific River Edge, NJ), 335-345 · Zbl 0832.34038
[10] Kaul, S. K., Stability and asymptotic stability in impulsive semidynamical systems, J. Appl. Math. Stoch. Anal., 7, 4, 509-523 (1994) · Zbl 0857.54039
[11] Kaul, S. K., Continuous and discrete flows and the property of \(ch 0^\pm \), Fund. Math., 127, 1-7 (1986) · Zbl 0613.58043
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