Differential Equations and Computational Simulations III
Electron. J. Diff. Eqns., Conf. 01, 1997, pp. 97-108.

On properties of nonlinear second order systems under nonlinear impulse perturbations

John R. Graef & Janos Karsai

Abstract:
In this paper, we consider the impulsive second order system
$$ \ddot{x}+f(x)=0\quad (t\neq t_{n});\quad \dot{x}(t_{n}+0)=b_{n}\dot{x}(t_{n}) \quad (t=t_{n}) $$
where $t_n=t_0+n\,p$ $(p>0, n=1,2\dots )$. In a previous paper, the authors proved that if $f(x)$ is strictly nonlinear, then this system has infinitely many periodic solutions. The impulses account for the main differences in the attractivity properties of the zero solution. Here, we prove that these periodic solutions are attractive in some sense, and we give good estimates for the attractivity region.

Published November 12, 1998.
Mathematics Subject Classifications: 34D05, 34D20, 34C15.
Key words and phrases: Asymptotic stability, attractrivity of periodic solutions, impulsive systems, nonlinear equations, second order systems.

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John R. Graef
Department of Mathematics and Statistics, Mississippi State University
Mississippi State, MS 39762 USA.
Email address: graef@math.msstate.edu

Janos Karsai
Department of Medical Informatics, Albert Szent-Gyorgyi Medical University
Szeged, Koranyi fasor 9, Hungary
Email address: karsai@silver.szote.u-szeged.hu

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