f2f8a01a5bac1bd3fc337488cc3fa1aa mat_a2024m9v12n18p2806.pdf bed28e2235db0f85ed2e0036f87beddca545944e mat_a2024m9v12n18p2806.pdf 98e52557b4045f3c16cdd865d494dfcf2bd3cdc9383fde4d212d87bbf3837fbf mat_a2024m9v12n18p2806.pdf Title: Rigid Polynomial Differential Systemswith Homogeneous Nonlinearities Subject: Planar differential systems whose angular velocity is constant are called rigid or uniform differential systems. The first rigid system goes back to the pendulum clock of Christiaan Huygens in 1656; since then, the interest for the rigid systems has been growing. Thus, at this moment, in MathSciNet there are 108 articles with the words rigid systems or uniform systems in their titles. Here, we study the dynamics of the planar rigid polynomial differential systems with homogeneous nonlinearities of arbitrary degree. More precisely, we characterize the existence and non-existence of limit cycles in this class of rigid systems, and we determine the local phase portraits of their finite and infinite equilibrium points in the Poincaré disc. Finally, we classify the global phase portraits in the Poincaré disc of the rigid polynomial differential systems of degree two, and of one class of rigid polynomial differential systems with cubic homogeneous nonlinearities that can exhibit one limit cycle. Keywords: quadratic systems; quadratic differential systems; limit cycles; rigid system; rigiddifferential systems Author: Jaume Llibre Creator: LaTeX with hyperref Producer: pdfTeX-1.40.25 CreationDate: Wed Sep 11 10:36:04 2024 CEST ModDate: Wed Sep 11 10:50:17 2024 CEST Custom Metadata: no Metadata Stream: no Tagged: no UserProperties: no Suspects: no Form: none JavaScript: no Pages: 10 Encrypted: no Page size: 595.276 x 841.89 pts (A4) Page rot: 0 File size: 720122 bytes Optimized: no PDF version: 1.7 name type encoding emb sub uni object ID ------------------------------------ ----------------- ---------------- --- --- --- --------- GEQIWM+URWPalladioL-Bold Type 1 Custom yes yes yes 10 0 BVKCCY+URWPalladioL-Roma Type 1 Custom yes yes yes 16 0 PLZQED+URWPalladioL-Ital Type 1 Custom yes yes yes 21 0 SOSTRQ+CMR10 Type 1 Builtin yes yes yes 26 0 LHZOBG+PazoMath-Italic Type 1 Builtin yes yes yes 31 0 SYFPBV+CMMI10 Type 1 Builtin yes yes yes 36 0 WBPHGE+CMSY10 Type 1 Builtin yes yes yes 41 0 XVOAMZ+MSBM10 Type 1 Builtin yes yes yes 46 0 DVTHKG+TimesNewRomanPS-BoldItalicMT Type 1C WinAnsi yes yes no 62 0 BEGLGA+CMEX10 Type 1 Builtin yes yes yes 79 0 QGSPAQ+PazoMath Type 1 Builtin yes yes yes 108 0 Jhove (Rel. 1.28.0, 2023-05-18) Date: 2024-11-27 02:49:10 CET RepresentationInformation: mat_a2024m9v12n18p2806.pdf ReportingModule: PDF-hul, Rel. 1.12.4 (2023-03-16) LastModified: 2024-11-26 18:43:25 CET Size: 720122 Format: PDF Version: 1.7 Status: Well-Formed and valid SignatureMatches: PDF-hul MIMEtype: application/pdf PDFMetadata: Objects: 276 FreeObjects: 1 IncrementalUpdates: 0 DocumentCatalog: PageLayout: SinglePage PageMode: UseNone Outlines: Item: Title: Introduction and Statement of the Main Result Destination: section.1 Item: Title: Proofs Destination: section.2 Item: Title: Phase Portraits Destination: section.3 Item: Title: Conclusions Destination: section.4 Item: Title: Appendix A Destination: appendix.A. Children: Item: Title: Appendix A.1 Destination: subsection.A.1 Item: Title: Appendix A.2 Destination: subsection.A.2 Item: Title: References Destination: appendix.B. Info: Title: Rigid Polynomial Differential Systemswith Homogeneous Nonlinearities Author: Jaume Llibre Subject: Planar differential systems whose angular velocity is constant are called rigid or uniform differential systems. The first rigid system goes back to the pendulum clock of Christiaan Huygens in 1656; since then, the interest for the rigid systems has been growing. Thus, at this moment, in MathSciNet there are 108 articles with the words rigid systems or uniform systems in their titles. Here, we study the dynamics of the planar rigid polynomial differential systems with homogeneous nonlinearities of arbitrary degree. More precisely, we characterize the existence and non-existence of limit cycles in this class of rigid systems, and we determine the local phase portraits of their finite and infinite equilibrium points in the Poincaré disc. Finally, we classify the global phase portraits in the Poincaré disc of the rigid polynomial differential systems of degree two, and of one class of rigid polynomial differential systems with cubic homogeneous nonlinearities that can exhibit one limit cycle. 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