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Lyapunov quasi-stable trajectories

Volume 220 / 2013

Changming Ding Fundamenta Mathematicae 220 (2013), 139-154 MSC: Primary 37B25; Secondary 54H20. DOI: 10.4064/fm220-2-4

Abstract

We introduce the notions of Lyapunov quasi-stability and Zhukovskiĭ quasi-stability of a trajectory in an impulsive semidynamical system defined in a metric space, which are counterparts of corresponding stabilities in the theory of dynamical systems. We initiate the study of fundamental properties of those quasi-stable trajectories, in particular, the structures of their positive limit sets. In fact, we prove that if a trajectory is asymptotically Lyapunov quasi-stable, then its limit set consists of rest points, and if a trajectory in a locally compact space is uniformly asymptotically Zhukovskiĭ quasi-stable, then its limit set is a rest point or a periodic orbit. Also, we present examples to show the differences between variant quasi-stabilities. Further, some sufficient conditions are given to guarantee the quasi-stabilities of a prescribed trajectory.

Authors

  • Changming DingSchool of Mathematical Sciences
    Xiamen University
    Xiamen, Fujian 361005, P.R. China
    e-mail

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