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Limit sets and the Poincaré-Bendixson theorem in impulsive semidynamical systems. (English) Zbl 1143.37014
An impulsive semidynamical system (X,π,Ω,M,I) consists of a semidynamical system π:X× + X on a metric space X, an open set Ω in X, a nonempty set M=Ω and a continuous function I:MΩ. The set M is called the impulsive set and I is called the impulse function. The impulsive trajectory π ˜(x,t) is obtained from π(x,t) and the action of I; when π(x,t) hits M, the motion continues at I(π(x,t)). The corresponding impulsive limit set of xX is denoted by L ˜ + (x). The authors study mainly the case X= 2 and prove, as their main result, that if Ω 2 has the compact closure, xΩ, and L ˜ + (x) admits neither rest points nor initial points, then L ˜ + (x) is a periodic orbit.
MSC:
37B99Topological dynamics
34C05Location of integral curves, singular points, limit cycles (ODE)
54H20Topological dynamics
34A37Differential equations with impulses
References:
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