ab8966a641570633eb443ac37f541738 mathematics-13-00756.pdf ef910012c048c32296ea8d9d01f087dfba5d655b mathematics-13-00756.pdf f282bd1178240a7f19fb9072c7a1f54a8ac037541ab8c8c8749ba114943ef89d mathematics-13-00756.pdf Title: The Global Dynamics of the Painlevé–Gambier Equations XVIII, XXI, and XXII Subject: In this paper, we describe the global dynamics of the Painlevé–Gambier equations numbered XVIII: x''-(x')2/(2x)-4x2=0, XXI: x''-3(x')2/(4x)-3x2, and XXII: x''-3(x')2/(4x)+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 Author: Jie Li and Jaume Llibre Creator: LaTeX with hyperref Producer: pdfTeX-1.40.25 CreationDate: Tue Feb 25 16:31:11 2025 CET ModDate: Tue Feb 25 16:59:00 2025 CET Custom Metadata: no Metadata Stream: no Tagged: no UserProperties: no Suspects: no Form: none JavaScript: no Pages: 12 Encrypted: no Page size: 595.276 x 841.89 pts (A4) Page rot: 0 File size: 1424903 bytes Optimized: no PDF version: 1.7 name type encoding emb sub uni object ID ------------------------------------ ----------------- ---------------- --- --- --- --------- OKTVOW+URWPalladioL-Roma Type 1 Custom yes yes yes 10 0 IJPMCU+URWPalladioL-Bold Type 1 Custom yes yes yes 16 0 SPGVPO+URWPalladioL-Ital Type 1 Custom yes yes yes 21 0 VGYPIE+CMSY10 Type 1 Builtin yes yes yes 26 0 GIGFZE+CMR10 Type 1 Builtin yes yes yes 31 0 QDTWCG+MSBM10 Type 1 Builtin yes yes yes 57 0 SYFPBV+CMMI10 Type 1 Builtin yes yes yes 65 0 DXJXEP+CMEX10 Type 1 Builtin yes yes yes 81 0 AXSMLR+PazoMath-Italic Type 1 Builtin yes yes yes 86 0 QHMKWQ+PazoMath Type 1 Builtin yes yes yes 94 0 Jhove (Rel. 1.28.0, 2023-05-18) Date: 2025-04-08 02:07:48 CEST RepresentationInformation: mathematics-13-00756.pdf ReportingModule: PDF-hul, Rel. 1.12.4 (2023-03-16) LastModified: 2025-04-07 11:43:44 CEST Size: 1424903 Format: PDF Version: 1.7 Status: Well-Formed and valid SignatureMatches: PDF-hul MIMEtype: application/pdf PDFMetadata: Objects: 303 FreeObjects: 1 IncrementalUpdates: 0 DocumentCatalog: PageLayout: SinglePage PageMode: UseNone Outlines: Item: Title: Introduction Destination: section.1 Item: Title: The Painlevé Gambier Equations XVIII and XXI Destination: section.2 Children: Item: Title: Finite Equilibrium Points Destination: subsection.2.1 Item: Title: Infinite Equilibrium Points Destination: subsection.2.2 Item: Title: Phase Portraits of Painlevé Gambier Equations XVIII and XXI Destination: subsection.2.3 Item: Title: The Painlevé Gambier Equation XXII Destination: section.3 Children: Item: Title: Finite Equilibrium Points Destination: subsection.3.1 Item: Title: Infinite Equilibrium Points Destination: subsection.3.2 Item: Title: Phase Portraits of Painlevé Gambier Equation XXII Destination: subsection.3.3 Item: Title: References Destination: section.4 Info: Title: The Global Dynamics of the Painlevé Gambier Equations XVIII, XXI, and XXII Author: Jie Li and Jaume Llibre Subject: In this paper, we describe the global dynamics of the Painlevé Gambier equations numbered XVIII: x''-(x')2/(2x)-4x2=0, XXI: x''-3(x')2/(4x)-3x2, and XXII: x''-3(x')2/(4x)+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. 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