Lyapunov quasi-stable trajectories

Changming Ding¹

¹ School of Mathematical Sciences Xiamen University Xiamen, Fujian 361005, P.R. China

Fundamenta Mathematicae, Tome 220 (2013) no. 2, pp. 139-154.

Résumé

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.

Détail

DOI : 10.4064/fm220-2-4

Keywords: introduce notions lyapunov quasi stability zhukovski quasi stability trajectory impulsive semidynamical system defined metric space which counterparts corresponding stabilities theory dynamical systems initiate study fundamental properties those quasi stable trajectories particular structures their positive limit sets prove trajectory asymptotically lyapunov quasi stable its limit set consists rest points trajectory locally compact space uniformly asymptotically zhukovski quasi stable its limit set rest point periodic orbit present examples differences between variant quasi stabilities further sufficient conditions given guarantee quasi stabilities prescribed trajectory