The Global Dynamics of the Painlevé-Gambier Equations XVIII, XXI, and XXII
Li, Jie (Southwest Petroleum University. School of Sciences)
Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)

Date:	2025
Abstract:	In this paper, we describe the global dynamics of the Painlevé-Gambier equations numbered XVIII: (Formula presented. ), XXI: (Formula presented. ), and XXII: (Formula presented. ). We obtain three rational functions as their first integrals and classify their phase portraits in the Poincaré disc. The main reason for considering these three Painlevé-Gambier equations is due to the paper of Guha, P. , et al. , where the authors studied these three differential equations in order to illustrate a method to generate nonlocal constants of motion for a special class of nonlinear differential equations. Here, we want to complete their studies describing all of the dynamics of these equations. This demonstrates that the phase portraits of equations XVIII and XXI in the Poincaré disc are topologically equivalent.
Grants:	European Commission 777911 Agencia Estatal de Investigación PID2022-136613NB-I00 Agència de Gestió d'Ajuts Universitaris i de Recerca 2021/SGR-00113
Rights:	Aquest document està subjecte a una llicència d'ús Creative Commons. Es permet la reproducció total o parcial, la distribució, la comunicació pública de l'obra i la creació d'obres derivades, fins i tot amb finalitats comercials, sempre i quan es reconegui l'autoria de l'obra original.
Language:	Anglès
Document:	Article ; recerca ; Versió publicada
Subject:	Painlevé-Gambier equations ; Phase portrait ; Poincaré disc ; First integral
Published in:	Mathematics, Vol. 13, Issue 5 (February 2025) , art. 756, ISSN 2227-7390

DOI: 10.3390/math13050756

12 p, 1.4 MB

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