b82a7b90a82d13811c4087a115e48379 sym_a2023m8v15n8p1476.pdf 2a4b2aa6349f642efd20d4be93c1a70c58d400aa sym_a2023m8v15n8p1476.pdf 9a822dcf5375d39a0eed17467c2df5d0638bd6ea47cc4c7aa2078d6a7ccdaa5a sym_a2023m8v15n8p1476.pdf Title: Symmetric Phase Portraits of Homogeneous Polynomial Hamiltonian Systems of Degree 1, 2, 3, 4, and 5 with Finitely Many Equilibria Subject: Roughly speaking, the Poincaré disc D2 is the closed disc centered at the origin of the coordinates of R2, where the whole of R2 is identified with the interior of D2 and the circle of the boundary of D2 is identified with the infinity of R2, because in the plane R2, we can go to infinity in as many directions as points have the circle. The phase portraits of the quadratic Hamiltonian systems in the Poincaré disc were classified in 1994. Since then, no new interesting classes of Hamiltonian systems have been classified on the Poincaré disc. In this paper, we determine the phase portraits in the Poincaré disc of five classes of homogeneous Hamiltonian polynomial differential systems of degrees 1, 2, 3, 4, and 5 with finitely many equilibria. Moreover, all these phase portraits are symmetric with respect to the origin of coordinates. We showed that these polynomial differential systems exhibit precisely 2, 2, 3, 3, and 4 topologically distinct phase portraits in the Poincaré disc. Of course, the new results are for the homogeneous Hamiltonian polynomial differential systems of degrees 3, 4, and 5. The tools used here for obtaining these phase portraits also work for obtaining any phase portrait of a homogeneous Hamiltonian polynomial differential system of arbitrary degree. Keywords: homogeneous Hamiltonian system; phase portrait; Poincaré disc Author: Rebiha Benterki and Jaume Llibre Creator: LaTeX with hyperref Producer: pdfTeX-1.40.21 CreationDate: Tue Jul 25 16:31:33 2023 CEST ModDate: Thu Mar 7 18:17:20 2024 CET 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: 982528 bytes Optimized: yes PDF version: 1.7 name type encoding emb sub uni object ID ------------------------------------ ----------------- ---------------- --- --- --- --------- XVOAMZ+MSBM10 Type 1 Builtin yes yes no 277 0 PDUCPO+URWPalladioL-Bold Type 1 Custom yes yes no 281 0 RPYMEG+URWPalladioL-Roma Type 1 Custom yes yes no 284 0 XCJKPN+URWPalladioL-Ital Type 1 Custom yes yes no 287 0 GIGFZE+CMR10 Type 1 Builtin yes yes no 290 0 DWFPMF+PazoMath-Italic Type 1 Builtin yes yes no 293 0 ZCVQXI+CMSY10 Type 1 Builtin yes yes no 296 0 GXBWHU+TimesNewRomanPS-BoldMT Type 1C WinAnsi yes yes no 305 0 CSRIGL+TimesNewRomanPS-BoldItalicMT Type 1C WinAnsi yes yes no 307 0 NIWPUW+CMEX10 Type 1 Builtin yes yes no 170 0 SYFPBV+CMMI10 Type 1 Builtin yes yes no 171 0 URYLRZ+URWPalladioL-BoldItal Type 1 Custom yes yes no 177 0 GGZKKH+CMBX10 Type 1 Builtin yes yes no 178 0 Jhove (Rel. 1.28.0, 2023-05-18) Date: 2024-03-08 02:36:37 CET RepresentationInformation: sym_a2023m8v15n8p1476.pdf ReportingModule: PDF-hul, Rel. 1.12.4 (2023-03-16) LastModified: 2024-03-07 18:17:38 CET Size: 982528 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 and Basic Results Destination: section.2 Children: Item: Title: Poincaré Compactification Destination: subsection.2.1 Item: Title: Phase Portraits on the Poincaré disc Destination: subsection.2.2 Item: Title: Homogeneous Polynomial Hamiltonian Systems Destination: subsection.2.3 Item: Title: Proof of Theorem 1 for n=1 Destination: section.3 Item: Title: Proof of Theorem 1 for n=2 Destination: section.4 Item: Title: Proof of Theorem 1 for n=3 Destination: section.5 Item: Title: Proof of Theorem 1 for n=4 Destination: section.6 Item: Title: Proof of Theorem 1 for n=5 Destination: section.7 Item: Title: Conclusions Destination: section.8 Item: Title: References Destination: section.9 Filters: FilterPipeline: FlateDecode Fonts: Type1: Font: BaseFont: URYLRZ+URWPalladioL-BoldItal FontSubset: true FirstChar: 110 LastChar: 110 FontDescriptor: FontName: URYLRZ+URWPalladioL-BoldItal Flags: Symbolic FontBBox: -170, -300, 1073, 935 FontFile: true EncodingDictionary: Differences: true Font: BaseFont: GIGFZE+CMR10 FontSubset: true FirstChar: 40 LastChar: 61 FontDescriptor: FontName: GIGFZE+CMR10 Flags: Symbolic FontBBox: -40, -250, 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NIWPUW+CMEX10 FontSubset: true FirstChar: 0 LastChar: 21 FontDescriptor: FontName: NIWPUW+CMEX10 Flags: Symbolic FontBBox: -24, -2960, 1454, 772 FontFile: true Font: BaseFont: SYFPBV+CMMI10 FontSubset: true FirstChar: 60 LastChar: 62 FontDescriptor: FontName: SYFPBV+CMMI10 Flags: Symbolic FontBBox: -32, -250, 1048, 750 FontFile: true Font: BaseFont: RPYMEG+URWPalladioL-Roma FontSubset: true FirstChar: 1 LastChar: 243 FontDescriptor: FontName: RPYMEG+URWPalladioL-Roma Flags: Symbolic FontBBox: -166, -283, 1021, 943 FontFile: true EncodingDictionary: Differences: true Font: BaseFont: XCJKPN+URWPalladioL-Ital FontSubset: true FirstChar: 2 LastChar: 233 FontDescriptor: FontName: XCJKPN+URWPalladioL-Ital Flags: Symbolic FontBBox: -170, -305, 1010, 941 FontFile: true EncodingDictionary: Differences: true XMP: application/pdf Rebiha Benterki and Jaume Llibre Roughly speaking, the Poincaré disc D2 is the closed disc centered at the origin of the coordinates of R2, where the whole of R2 is identified with the interior of D2 and the circle of the boundary of D2 is identified with the infinity of R2, because in the plane R2, we can go to infinity in as many directions as points have the circle. The phase portraits of the quadratic Hamiltonian systems in the Poincaré disc were classified in 1994. Since then, no new interesting classes of Hamiltonian systems have been classified on the Poincaré disc. In this paper, we determine the phase portraits in the Poincaré disc of five classes of homogeneous Hamiltonian polynomial differential systems of degrees 1, 2, 3, 4, and 5 with finitely many equilibria. Moreover, all these phase portraits are symmetric with respect to the origin of coordinates. We showed that these polynomial differential systems exhibit precisely 2, 2, 3, 3, and 4 topologically distinct phase portraits in the Poincaré disc. Of course, the new results are for the homogeneous Hamiltonian polynomial differential systems of degrees 3, 4, and 5. The tools used here for obtaining these phase portraits also work for obtaining any phase portrait of a homogeneous Hamiltonian polynomial differential system of arbitrary degree. Symmetric Phase Portraits of Homogeneous Polynomial Hamiltonian Systems of Degree 1, 2, 3, 4, and 5 with Finitely Many Equilibria 2023-07-25T16:31:33+02:00 LaTeX with hyperref 2024-03-07T18:17:20+01:00 2024-03-07T18:17:20+01:00 homogeneous Hamiltonian system; phase portrait; Poincaré disc pdfTeX-1.40.21 uuid:5fdd8efa-dfd8-4ea5-bebe-2794bd75fe58 uuid:23d7458c-b7c8-43f8-bd17-66492ca7441e Pages: Page: Sequence: 1 Annotations: Annotation: Subtype: Link Rect: 34, 281, 150, 290 Annotation: Subtype: Link Rect: 34, 164, 96, 186 Annotation: Subtype: Link Rect: 126, 90, 154, 99 Annotation: Subtype: Link Rect: 34, 78, 147, 87 Annotation: Subtype: Link Rect: 34, 756, 177, 792 Annotation: Subtype: Link Rect: 515, 756, 560, 792 Annotation: Subtype: Link Rect: 121, 642, 132, 655 Annotation: Subtype: Link Rect: 231, 642, 242, 655 Annotation: Subtype: Link Rect: 405, 238, 412, 247 ActionDest: 38 Annotation: Subtype: Link Rect: 415, 238, 422, 247 ActionDest: 38 Annotation: Subtype: Link Rect: 523, 238, 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