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Limit sets and the Poincaré-Bendixson theorem in impulsive semidynamical systems. (English) Zbl 1143.37014
An impulsive semidynamical system $\left(X,\pi ,{\Omega },M,I\right)$ consists of a semidynamical system $\pi :X×{ℝ}^{+}\to X$ on a metric space $X$, an open set ${\Omega }$ in $X$, a nonempty set $M=\partial {\Omega }$ and a continuous function $I:M\to {\Omega }$. The set $M$ is called the impulsive set and $I$ is called the impulse function. The impulsive trajectory $\stackrel{˜}{\pi }\left(x,t\right)$ is obtained from $\pi \left(x,t\right)$ and the action of $I$; when $\pi \left(x,t\right)$ hits $M$, the motion continues at $I\left(\pi \left(x,t\right)\right)$. The corresponding impulsive limit set of $x\in X$ is denoted by ${\stackrel{˜}{L}}^{+}\left(x\right)$. The authors study mainly the case $X={ℝ}^{2}$ and prove, as their main result, that if ${\Omega }\subseteq {ℝ}^{2}$ has the compact closure, $x\in {\Omega }$, and ${\stackrel{˜}{L}}^{+}\left(x\right)$ admits neither rest points nor initial points, then ${\stackrel{˜}{L}}^{+}\left(x\right)$ is a periodic orbit.
MSC:
 37B99 Topological dynamics 34C05 Location of integral curves, singular points, limit cycles (ODE) 54H20 Topological dynamics 34A37 Differential equations with impulses
References:
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