The Global Dynamics of the Painlevé–Gambier Equations XVIII, XXI, and XXII

Jie Li and Jaume Llibre ()
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Jie Li: School of Sciences, Southwest Petroleum University, 610500 Chengdu, China
Jaume Llibre: Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra, 08193 Barcelona, Catalonia, Spain

Mathematics, 2025, vol. 13, issue 5, 1-12

Abstract: In this paper, we describe the global dynamics of the Painlevé–Gambier equations numbered XVIII: x ″ − ( x ′ ) 2 / ( 2 x ) − 4 x 2 = 0 , XXI: x ″ − 3 ( x ′ ) 2 / ( 4 x ) − 3 x 2 , and XXII: x ″ − 3 ( x ′ ) 2 / ( 4 x ) + 1 = 0 . 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.

Keywords: Painlevé–Gambier equations; phase portrait; Poincaré disc; first integral (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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