70720b5f0cac3181ed9f2b261a1700ae axi_a2025m4v14n4p321.pdf 2bb2ebff3536e51b6b38166215e28621c76bc228 axi_a2025m4v14n4p321.pdf 2578895d81ceba2c611701bd17d2d82c9616540eb9b521e34f35bf00cb2bdd38 axi_a2025m4v14n4p321.pdf Title: On the Dynamics of a Modified van der Pol–Duffing Oscillator Subject: The 3-dimensional modified van der Pol–Duffing oscillator has been studied by several authors. We complete its study, first characterizing its zero-Hopf blackequilibria and then its zero-Hopf bifurcations—i.e., we provide sufficient conditions for the existence of three, two or one periodic solutions, bifurcating from the zero-Hopf equilibrium localized at the origin of coordinates. Recall that an equilibrium point blackof a 3-dimensional differential system whose eigenvalues are zero and a pair of purely imaginary eigenvalues is a zero-Hopf equilibrium. blackFinally, we determine the dynamics of this system near infinity, i.e., we control the orbits that escape to or come from the infinity. Keywords: van der Pol–Duffing oscillator; zero-Hopf; dynamics at infinity Author: Firstname Lastname, Firstname Lastname and Firstname Lastname Creator: LaTeX with hyperref Producer: pdfTeX-1.40.25 CreationDate: Mon Apr 21 11:48:24 2025 CEST ModDate: Wed Jul 9 15:16:54 2025 CEST Custom Metadata: no Metadata Stream: yes Tagged: no UserProperties: no Suspects: no Form: none JavaScript: no Pages: 13 Encrypted: no Page size: 595.276 x 841.89 pts (A4) Page rot: 0 File size: 328753 bytes Optimized: yes PDF version: 1.7 name type encoding emb sub uni object ID ------------------------------------ ----------------- ---------------- --- --- --- --------- GBLQPF+CMEX10 Type 1 Builtin yes yes yes 298 0 JIEVJJ+URWPalladioL-Roma Type 1 Custom yes yes yes 302 0 YNWTNA+URWPalladioL-Bold Type 1 Custom yes yes yes 305 0 DQHRYX+URWPalladioL-Ital Type 1 Custom yes yes yes 308 0 OCFDXW+PazoMath-Italic Type 1 Builtin yes yes yes 311 0 SOSTRQ+CMR10 Type 1 Builtin yes yes yes 314 0 VKDSEL+CMSY10 Type 1 Builtin yes yes yes 317 0 MLJUUK+MSBM10 Type 1 Builtin yes yes yes 320 0 SYFPBV+CMMI10 Type 1 Builtin yes yes yes 208 0 SOJMCF+PazoMath Type 1 Builtin yes yes yes 209 0 Jhove (Rel. 1.28.0, 2023-05-18) Date: 2025-07-10 02:07:58 CEST RepresentationInformation: axi_a2025m4v14n4p321.pdf ReportingModule: PDF-hul, Rel. 1.12.4 (2023-03-16) LastModified: 2025-07-09 15:17:13 CEST Size: 328753 Format: PDF Version: 1.7 Status: Well-Formed and valid SignatureMatches: PDF-hul MIMEtype: application/pdf Profile: Linearized PDF PDFMetadata: Objects: 0 FreeObjects: 1 IncrementalUpdates: 0 DocumentCatalog: PageLayout: SinglePage PageMode: UseNone Outlines: Item: Title: Introduction and Statement of the Main Results Destination: section.1 Item: Title: Preliminaries Destination: section.2 Children: Item: Title: The Averaging Theory of First and Second Order Destination: subsection.2.1 Item: Title: Poincaré Compactification in R3 Destination: subsection.2.2 Item: Title: Proofs of Propositions 1 and 2 Destination: section.3 Item: Title: Proof of Theorem 1 Destination: section.4 Item: Title: Proof of Theorem 2 Destination: section.5 Item: Title: blackConclusions Destination: section.6 Item: Title: Appendix A Destination: appendix.A. Item: Title: References Destination: appendix.B. 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We complete its study, first characterizing its zero-Hopf blackequilibria and then its zero-Hopf bifurcations—i.e., we provide sufficient conditions for the existence of three, two or one periodic solutions, bifurcating from the zero-Hopf equilibrium localized at the origin of coordinates. Recall that an equilibrium point blackof a 3-dimensional differential system whose eigenvalues are zero and a pair of purely imaginary eigenvalues is a zero-Hopf equilibrium. blackFinally, we determine the dynamics of this system near infinity, i.e., we control the orbits that escape to or come from the infinity. 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