On the index of an isolated equilibrium point for piecewise polynomial differential systems
Li, Shimin (Hangzhou Normal University. School of Mathematics)
Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Zhao, Yulin (Sun Yat-sen University. School of Mathematics)

Date:	2025
Abstract:	The index of an equilibrium point is an integer that characterizes the topological structure of the equilibrium. There are two primary methods for calculating the index of an isolated equilibrium point in smooth differential systems. One approach, derived by Poincaré and Bendixson, states that the index can be determined by the number of hyperbolic and elliptic sectors in a neighborhood of the equilibrium point. This is known as the Poincaré-Bendixson formula. Alternatively, various studies have proposed an algebraic method for computing the index, often referred to as Cauchy index. In this paper, we extend Cauchy index method to calculate the index of equilibrium points in piecewise polynomial differential systems. We also apply these results to piecewise smooth quadratic quasi-homogeneous differential systems.
Grants:	Agencia Estatal de Investigación PID2019-104658GB-I00 European Commission 777911 Agència de Gestió d'Ajuts Universitaris i de Recerca 2021/SGR-00113
Note:	Altres ajuts: Reial Acadèmia de Ciències i Arts de Barcelona
Rights:	Aquest document està subjecte a una llicència d'ús Creative Commons. Es permet la reproducció total o parcial, la distribució, i la comunicació pública de l'obra, sempre que no sigui amb finalitats comercials, i sempre que es reconegui l'autoria de l'obra original. No es permet la creació d'obres derivades.
Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Cauchy index ; Piecewise polynomial differential systems ; Index of an equilibrium point
Published in:	Journal of mathematical analysis and applications, Vol. 551, Issue 2 Part 1 (November 2025) , art. 129651, ISSN 1096-0813

DOI: 10.1016/j.jmaa.2025.129651

Available from: 2027-11-30 Postprint

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Research literature > UAB research groups literature > Research Centres and Groups (research output) > Experimental sciences > GSD (Dynamical systems)
Articles > Research articles
Articles > Published articles