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Unbounded solutions of positively damped Liénard equations

100%
Ding C.
Annales Polonici Mathematici
|
1996
|
tom 64
|
nr 2
115-120
EN
This paper discusses the asymptotic behavior of solutions of the Liénard equation, especially the global behavior of unbounded solutions, and also gives a class of sufficient and necessary conditions for the orbit of a solution to intersect the vertical isocline.
2
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Lyapunov quasi-stable trajectories

100%
Ding C.
Fundamenta Mathematicae
|
2013
|
tom 220
|
nr 2
139-154
EN
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.
3
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A predator-prey model with state dependent impulsive effects

100%
Ding C.
Annales Polonici Mathematici
|
2014
|
tom 111
|
nr 3
297-308
EN
We investigate a Lotka-Volterra predator-prey model with state dependent impulsive effects, in which the control strategies by releasing natural enemies and spraying pesticide at different thresholds are considered. We present some sufficient conditions to guarantee the existence and asymptotical stability of semi-trivial periodic solutions and positive periodic solutions.
4
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Prolongational centers and their depths

63%
Ding B., Ding C.
Fundamenta Mathematicae
|
2016
|
tom 234
|
nr 3
287-296
EN
In 1926 Birkhoff defined the center depth, one of the fundamental invariants that characterize the topological structure of a dynamical system. In this paper, we introduce the concepts of prolongational centers and their depths, which lead to a complete family of topological invariants. Some basic properties of the prolongational centers and their depths are established. Also, we construct a dynamical system in which the depth of a prolongational center is a prescribed countable ordinal.
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