af4ab89ded634b832bcb151f132acd4c axioms-13-00161.pdf 316147c342fff7a9e42b01a35fb10132e67a506e axioms-13-00161.pdf 83f16d647c9589b586ea03d0c259038409cff3b5045793cad35d3e65078bd1a2 axioms-13-00161.pdf Title: Limit Cycles of Discontinuous Piecewise Differential Hamiltonian Systems Separated by a Straight Line Subject: In this article, we study the maximum number of limit cycles of discontinuous piecewise differential systems, formed by two Hamiltonians systems separated by a straight line. We consider three cases, when both Hamiltonians systems in each side of the discontinuity line have simultaneously degree one, two or three. We obtain that in these three cases, this maximum number is zero, one and three, respectively. Moreover, we prove that there are discontinuous piecewise differential systems realizing these maximum number of limit cycles. Note that we have solved the extension of the 16th Hilbert problem about the maximum number of limit cycles that these three classes of discontinuous piecewise differential systems separated by one straight line and formed by two Hamiltonian systems with a degree either one, two, or three, which such systems can exhibit. Keywords: discontinuous piecewise differential systems; limit cycle; Hamiltonian systems Author: Joyce A. Casimiro and Jaume Llibre Creator: LaTeX with hyperref Producer: pdfTeX-1.40.25 CreationDate: Thu Feb 29 08:37:24 2024 CET ModDate: Thu Feb 29 08:52:04 2024 CET Custom Metadata: no Metadata Stream: no Tagged: no UserProperties: no Suspects: no Form: none JavaScript: no Pages: 7 Encrypted: no Page size: 595.276 x 841.89 pts (A4) Page rot: 0 File size: 248031 bytes Optimized: no PDF version: 1.7 name type encoding emb sub uni object ID ------------------------------------ ----------------- ---------------- --- --- --- --------- EEUWZM+URWPalladioL-Bold Type 1 Custom yes yes yes 10 0 DYCAJW+URWPalladioL-Roma Type 1 Custom yes yes yes 16 0 LSWCBS+URWPalladioL-Ital Type 1 Custom yes yes yes 21 0 SKSVGJ+TimesNewRomanPS-BoldItalicMT Type 1C WinAnsi yes yes no 37 0 MJCSKH+CMEX10 Type 1 Builtin yes yes yes 54 0 RHFXIK+PazoMath-Italic Type 1 Builtin yes yes yes 59 0 SOSTRQ+CMR10 Type 1 Builtin yes yes yes 64 0 SYIROE+CMSY10 Type 1 Builtin yes yes yes 69 0 BVTOFD+CMBX10 Type 1 Builtin yes yes yes 74 0 FGELNC+URWPalladioL-BoldItal Type 1 Custom yes yes yes 79 0 QDTWCG+MSBM10 Type 1 Builtin yes yes yes 87 0 Jhove (Rel. 1.28.0, 2023-05-18) Date: 2024-05-15 02:41:44 CEST RepresentationInformation: axioms-13-00161.pdf ReportingModule: PDF-hul, Rel. 1.12.4 (2023-03-16) LastModified: 2024-05-14 10:38:55 CEST Size: 248031 Format: PDF Version: 1.7 Status: Well-Formed and valid SignatureMatches: PDF-hul MIMEtype: application/pdf PDFMetadata: Objects: 204 FreeObjects: 1 IncrementalUpdates: 0 DocumentCatalog: PageLayout: SinglePage PageMode: UseNone Outlines: Item: Title: Introduction and Statement of the Main Results Destination: section.1 Item: Title: Proof of Theorem 1 Destination: section.2 Item: Title: Examples Destination: section.3 Item: Title: References Destination: section.4 Info: Title: Limit Cycles of Discontinuous Piecewise Differential Hamiltonian Systems Separated by a Straight Line Author: Joyce A. Casimiro and Jaume Llibre Subject: In this article, we study the maximum number of limit cycles of discontinuous piecewise differential systems, formed by two Hamiltonians systems separated by a straight line. We consider three cases, when both Hamiltonians systems in each side of the discontinuity line have simultaneously degree one, two or three. We obtain that in these three cases, this maximum number is zero, one and three, respectively. Moreover, we prove that there are discontinuous piecewise differential systems realizing these maximum number of limit cycles. Note that we have solved the extension of the 16th Hilbert problem about the maximum number of limit cycles that these three classes of discontinuous piecewise differential systems separated by one straight line and formed by two Hamiltonian systems with a degree either one, two, or three, which such systems can exhibit. Keywords: discontinuous piecewise differential systems; limit cycle; Hamiltonian systems Creator: LaTeX with hyperref Producer: pdfTeX-1.40.25 CreationDate: Thu Feb 29 08:37:24 CET 2024 ModDate: Thu Feb 29 08:52:04 CET 2024 ID: 0xc566e63ceac8542aa0e7b816439cdd79, 0xc566e63ceac8542aa0e7b816439cdd79 Filters: FilterPipeline: FlateDecode Fonts: Type1: Font: BaseFont: DYCAJW+URWPalladioL-Roma FontSubset: true FirstChar: 1 LastChar: 243 FontDescriptor: FontName: DYCAJW+URWPalladioL-Roma Flags: Symbolic FontBBox: -166, -283, 1021, 943 FontFile: true EncodingDictionary: Differences: true ToUnicode: true Font: BaseFont: SOSTRQ+CMR10 FontSubset: true FirstChar: 40 LastChar: 93 FontDescriptor: FontName: SOSTRQ+CMR10 Flags: Symbolic FontBBox: -40, -250, 1009, 750 FontFile: true ToUnicode: true Font: BaseFont: LSWCBS+URWPalladioL-Ital FontSubset: true FirstChar: 2 LastChar: 122 FontDescriptor: FontName: LSWCBS+URWPalladioL-Ital Flags: Symbolic FontBBox: -170, -305, 1010, 941 FontFile: true EncodingDictionary: Differences: true ToUnicode: true Font: BaseFont: SYIROE+CMSY10 FontSubset: true FirstChar: 0 LastChar: 112 FontDescriptor: FontName: SYIROE+CMSY10 Flags: Symbolic FontBBox: -29, -960, 1116, 775 FontFile: true ToUnicode: true Font: BaseFont: MJCSKH+CMEX10 FontSubset: true FirstChar: 0 LastChar: 26 FontDescriptor: FontName: MJCSKH+CMEX10 Flags: Symbolic FontBBox: -24, -2960, 1454, 772 FontFile: true ToUnicode: true Font: BaseFont: QDTWCG+MSBM10 FontSubset: true FirstChar: 82 LastChar: 82 FontDescriptor: FontName: QDTWCG+MSBM10 Flags: Symbolic FontBBox: -55, -420, 2343, 920 FontFile: true ToUnicode: true Font: BaseFont: EEUWZM+URWPalladioL-Bold FontSubset: true FirstChar: 3 LastChar: 134 FontDescriptor: FontName: EEUWZM+URWPalladioL-Bold Flags: Symbolic FontBBox: -152, -301, 1000, 935 FontFile: true EncodingDictionary: Differences: true ToUnicode: true Font: BaseFont: BVTOFD+CMBX10 FontSubset: true FirstChar: 40 LastChar: 41 FontDescriptor: FontName: BVTOFD+CMBX10 Flags: Symbolic FontBBox: -56, -250, 1164, 750 FontFile: true ToUnicode: true Font: BaseFont: RHFXIK+PazoMath-Italic FontSubset: true FirstChar: 182 LastChar: 182 FontDescriptor: FontName: RHFXIK+PazoMath-Italic Flags: Symbolic FontBBox: -70, -277, 902, 733 FontFile: true ToUnicode: true Font: BaseFont: FGELNC+URWPalladioL-BoldItal FontSubset: true FirstChar: 97 LastChar: 99 FontDescriptor: FontName: FGELNC+URWPalladioL-BoldItal Flags: Symbolic FontBBox: -170, -300, 1073, 935 FontFile: true EncodingDictionary: Differences: true ToUnicode: true Pages: Page: Sequence: 1 Annotations: Annotation: Subtype: Link Rect: 100, 293, 152, 302 Annotation: Subtype: Link Rect: 34, 282, 113, 290 Annotation: Subtype: Link Rect: 34, 164, 96, 186 Annotation: Subtype: Link Rect: 125, 90, 154, 99 Annotation: Subtype: Link Rect: 34, 78, 149, 87 Annotation: Subtype: Link Rect: 34, 756, 155, 792 Annotation: Subtype: Link Rect: 515, 756, 560, 792 Annotation: Subtype: Link Rect: 128, 660, 139, 674 Annotation: Subtype: Link Rect: 237, 660, 248, 674 Annotation: Subtype: Link Rect: 491, 331, 498, 340 ActionDest: 101 Annotation: Subtype: Link Rect: 501, 330, 508, 340 ActionDest: 101 Annotation: Subtype: Link Rect: 289, 293, 296, 302 ActionDest: 101 Annotation: Subtype: Link Rect: 385, 293, 392, 302 ActionDest: 101 Annotation: Subtype: Link Rect: 197, 280, 204, 289 ActionDest: 101 Annotation: Subtype: Link Rect: 201, 193, 208, 201 ActionDest: 101 Annotation: Subtype: Link Rect: 234, 180, 241, 189 ActionDest: 101 Annotation: Subtype: Link Rect: 461, 180, 468, 189 ActionDest: 101 Annotation: Subtype: Link Rect: 291, 167, 303, 176 ActionDest: 101 Annotation: Subtype: Link Rect: 545, 167, 557, 176 ActionDest: 101 Annotation: Subtype: Link Rect: 367, 155, 379, 164 ActionDest: 101 Annotation: Subtype: Link Rect: 491, 142, 503, 151 ActionDest: 107 Annotation: Subtype: Link Rect: 504, 142, 516, 151 ActionDest: 107 Annotation: Subtype: Link Rect: 392, 130, 404, 139 ActionDest: 107 Annotation: Subtype: Link Rect: 407, 130, 419, 139 ActionDest: 107 Annotation: Subtype: Link Rect: 109, 17, 259, 28 Annotation: Subtype: Link Rect: 411, 17, 560, 28 Annotation: Subtype: Link Rect: 34, 354, 87, 373 Page: Sequence: 2 Annotations: Annotation: Subtype: Link Rect: 504, 603, 516, 612 ActionDest: 107 Annotation: Subtype: Link Rect: 451, 552, 458, 563 ActionDest: 51 Annotation: Subtype: Link Rect: 259, 524, 265, 535 ActionDest: 51 Annotation: Subtype: Link Rect: 259, 511, 265, 522 ActionDest: 51 Annotation: Subtype: Link Rect: 259, 498, 265, 510 ActionDest: 51 Annotation: Subtype: Link Rect: 329, 483, 336, 494 ActionDest: 51 Annotation: Subtype: Link Rect: 228, 433, 235, 445 ActionDest: 51 Annotation: Subtype: Link Rect: 325, 433, 332, 445 ActionDest: 51 Annotation: Subtype: Link Rect: 258, 410, 265, 422 ActionDest: 51 Annotation: Subtype: Link Rect: 325, 395, 332, 407 ActionDest: 51 Annotation: Subtype: Link Rect: 433, 277, 440, 288 ActionDest: 51 Annotation: Subtype: Link Rect: 466, 277, 473, 288 ActionDest: 51 Annotation: Subtype: Link Rect: 259, 196, 266, 207 ActionDest: 51 Annotation: Subtype: Link Rect: 201, 93, 208, 104 ActionDest: 51 Page: Sequence: 3 Annotations: Annotation: Subtype: Link Rect: 371, 741, 378, 753 ActionDest: 51 Annotation: Subtype: Link Rect: 304, 731, 316, 740 ActionDest: 107 Annotation: Subtype: Link Rect: 324, 654, 331, 667 ActionDest: 51 Annotation: Subtype: Link Rect: 434, 493, 441, 504 ActionDest: 84 Annotation: Subtype: Link Rect: 467, 493, 474, 504 ActionDest: 84 Annotation: Subtype: Link Rect: 237, 410, 244, 421 ActionDest: 84 Annotation: Subtype: Link Rect: 442, 230, 449, 242 ActionDest: 92 Annotation: Subtype: Link Rect: 323, 210, 330, 222 ActionDest: 51 Page: Sequence: 4 Annotations: Annotation: Subtype: Link Rect: 550, 675, 557, 686 ActionDest: 84 Annotation: Subtype: Link Rect: 187, 663, 199, 674 ActionDest: 92 Annotation: Subtype: Link Rect: 440, 533, 452, 545 ActionDest: 92 Annotation: Subtype: Link Rect: 199, 321, 206, 331 ActionDest: 92 Annotation: Subtype: Link Rect: 528, 233, 535, 243 ActionDest: 51 Page: Sequence: 5 Annotations: Annotation: Subtype: Link Rect: 476, 741, 488, 753 ActionDest: 92 Annotation: Subtype: Link Rect: 513, 741, 525, 753 ActionDest: 92 Annotation: Subtype: Link Rect: 438, 729, 450, 740 ActionDest: 92 Annotation: Subtype: Link Rect: 200, 654, 212, 666 ActionDest: 95 Annotation: Subtype: Link Rect: 290, 582, 297, 594 ActionDest: 95 Annotation: Subtype: Link Rect: 196, 389, 207, 399 ActionDest: 92 Annotation: Subtype: Link Rect: 508, 389, 519, 399 ActionDest: 95 Annotation: Subtype: Link Rect: 409, 189, 421, 200 ActionDest: 95 Annotation: Subtype: Link Rect: 448, 189, 460, 200 ActionDest: 95 Annotation: Subtype: Link Rect: 376, 176, 388, 187 ActionDest: 92 Annotation: Subtype: Link Rect: 395, 89, 402, 102 ActionDest: 101 Page: Sequence: 6 Annotations: Annotation: Subtype: Link Rect: 221, 508, 232, 519 ActionDest: 95 Annotation: Subtype: Link Rect: 509, 508, 520, 519 ActionDest: 95 Annotation: Subtype: Link Rect: 240, 298, 277, 308 Annotation: Subtype: Link Rect: 283, 298, 319, 308 Annotation: Subtype: Link Rect: 458, 286, 495, 297 Annotation: Subtype: Link Rect: 501, 286, 537, 297 Annotation: Subtype: Link Rect: 100, 229, 137, 239 Annotation: Subtype: Link Rect: 238, 171, 275, 182 Annotation: Subtype: Link Rect: 104, 148, 142, 159 Annotation: Subtype: Link Rect: 301, 125, 338, 136 Annotation: Subtype: Link Rect: 155, 102, 193, 113 Annotation: Subtype: Link Rect: 383, 79, 420, 90 Page: Sequence: 7 Annotations: Annotation: Subtype: Link Rect: 149, 730, 186, 741 Annotation: Subtype: Link Rect: 327, 706, 365, 718 Annotation: Subtype: Link Rect: 131, 684, 168, 694 Annotation: Subtype: Link Rect: 192, 661, 229, 671 Annotation: Subtype: Link Rect: 118, 638, 155, 648 Annotation: Subtype: Link Rect: 255, 614, 293, 625 Annotation: Subtype: Link Rect: 229, 591, 266, 602 Annotation: Subtype: Link Rect: 181, 568, 219, 579 Annotation: Subtype: Link Rect: 104, 546, 141, 556 Checksum: 9a836cc4 Type: CRC32 Checksum: af4ab89ded634b832bcb151f132acd4c Type: MD5 Checksum: 316147c342fff7a9e42b01a35fb10132e67a506e Type: SHA-1 Checksum: 83f16d647c9589b586ea03d0c259038409cff3b5045793cad35d3e65078bd1a2 Type: SHA-256