A probabilistic approach of the Poincar{\'e}-Bendixon's problem in $\mathbb{R}^d$

TL;DR

This paper introduces a probabilistic model to analyze the long-term behavior of iterative systems and ODEs in higher-dimensional spaces, providing new insights into the Poincaré-Bendixson problem.

Contribution

It proposes a novel probabilistic framework for understanding asymptotic dynamics in $\

Findings

Probabilistic model effectively describes asymptotic behavior.

Application to ODEs offers new perspectives on Poincaré-Bendixson's problem.

Framework extends analysis to higher-dimensional spaces.

Abstract

We present how a probabilistic model can describe the asymptotic behaviour of the iterations, especially for ODE with an approach of the Poincar\'e-Bendixon's problem in $\mathbb{R}^d$. On pr\'esente un mod\`ele probabiliste pour d\'ecrire le comportement asymptotique d'une it{\'e}ration, en particulier pour les EDO et pour aborder le probl{\`e}me de Poincar\'e-Bendixon's dans $\mathbb{R}^d$.