Generalized recurrence in impulsive semidynamical systems
Keywords
Impulsive semidynamical system, generalized recurrence, Lyapunov function, higher prolongationAbstract
We aim to introduce the generalized recurrence into the theory of impulsive semidynamical systems. Similarly to Auslander's construction in [J. Auslander, {\it Generalized recurrence in dynamical systems}, Contrib. Differential Equations \textbf{3} (1964), 65-74], we present two different characterizations, respectively, by Lyapunov functions and higher prolongations. In fact, we show that if the phase space is a locally compact separable metric space, then the generalized recurrent set is the same as the quasi prolongational recurrent set. Also, we see that many new phenomena appear for the impulse effects in the semidynamical system.References
J. Auslander, Generalized recurrence in dynamical systems, Contrib. Differential Equations 3 (1964), 65–74.
N.P. Bhatia and O. Hajek, Local Semi-Dynamical Systems, Lecture Notes in Mathematics, vol. 90, Springer, Berlin, 1970.
N.P. Bhatia and G.P. Szegő, Stability Theory of Dynamical Systems, Springer, Berlin, 2002.
K. Ciesielski, On semicontinuity in impulsive systems, Bull. Polish Acad. Sci. Math. 52 (2004), 71–80.
K. Ciesielski, On stability in impulsive dynamical systems, Bull. Polish Acad. Sci. Math. 52 (2004), 81–91.
K. Ciesielski, On time reparametrizations and isomorphisms of impulsive dynamical systems, Ann. Polon. Math. 84 (2004), 1–25.
C. Ding, Lyapunov quasi-stable trajectories, Fund. Math. 220 (2013), 139–154.
C. Ding, Limit sets in impulsive semidynamical systems, Topol. Methods Nonlinear Anal. 43 (2014), 97–115.
B. Ding and C. Ding, Prolongational centers and their depths, Fund. Math. 234 (2016), 287–296.
J. Dugundji, Topology, Allyn and Bacon, 1966.
S.K. Kaul, On impulsive semidynamical systems, J. Math. Anal. Appl. 150 (1990), 120–128.
S.K. Kaul, On impulsive semidynamical systems II. Recursive properties, Nonlinear Anal. 16 (1991), 635–645.
S.K. Kaul, Stability and asymptotic stability in impulsive semidynamical systems, J. Appl. Math. Stochastic Anal. 7 (1991), 509–523.
V.V. Nemytskiı̆ and V.V. Stepanov, Qualitative Theory of Differential Equations, Princeton University Press, Princeton, 1966.
Z. Nitecki, Recurrent structure of completely unstable flows on surfaces of finite Euler characteristic, Amer. J. Math. 103 (1981), 143–180.
M.L.A. Peixoto, Characterizing Ω-stability for flows in the plane, Proc. Amer. Math. Soc. 104 (1988), 981–984.
S. Warner, The topology of compact convergence on continuous function spaces, Duke Math. J. 25 (1958), 265–282.
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 0
Number of citations: 0