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arXiv · 2411.01181

On the dynamics of non-autonomous systems in a~neighborhood of a~homoclinic trajectory

Alessandro Calamai·Matteo Franca·Michal Pospisil
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Abstract

This article is devoted to the study of a $2$-dimensional piecewise smooth (but possibly) discontinuous dynamical system, subject to a non-autonomous perturbation; we assume that the unperturbed system admits a homoclinic trajectory $\vec{\gamma}(t)$. Our aim is to analyze the dynamics in a neighborhood of $\vec{\gamma}(t)$ as the perturbation is turned on, by defining a Poincar\'e map and evaluating fly time and space displacement of trajectories performing a loop close to $\vec{\gamma}(t)$. Besides their intrinsic mathematical interest, these results can be thought of as a first step in the analysis of several interesting problems, such as the stability of a homoclinic trajectory of a non-autonomous ODE and a possible extension of Melnikov chaos to a discontinuous setting.

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Alessandro Calamai, Matteo Franca, Michal Pospisil. 2024-11-02. On the dynamics of non-autonomous systems in a~neighborhood of a~homoclinic trajectory. https://arxiv.org/abs/2411.01181

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