6fab5eb21689ed496ee10ca0f74f37ac symmetry_a2021v13n7p1128.pdf c1b1156c9447e7e55fef1904b9dbfc675c2445df symmetry_a2021v13n7p1128.pdf f3469d9f5788cbe791c50d1fcac8813519024b6e3a46bafb0045748c2dcc29db symmetry_a2021v13n7p1128.pdf Title: Limit Cycles of Planar Piecewise Differential Systems with Linear Hamiltonian Saddles Subject: We provide the maximum number of limit cycles for continuous and discontinuous planar piecewise differential systems formed by linear Hamiltonian saddles and separated either by one or two parallel straight lines. We show that when these piecewise differential systems are either continuous or discontinuous and are separated by one straight line, or are continuous and are separated by two parallel straight lines, they do not have limit cycles. On the other hand, when these systems are discontinuous and separated by two parallel straight lines, we prove that the maximum number of limit cycles that they can have is one and that this maximum is reached by providing an example of such a system with one limit cycle. When the line of discontinuity of the piecewise differential system is formed by one straight line, the symmetry of the problem allows to take this straight line without loss of generality as the line x=0. Similarly, when the line of discontinuity of the piecewise differential system is formed by two parallel straight lines due to the symmetry of the problem, we can assume without loss of generality that these two straight lines are x=1. Keywords: crossing limit cycles; linear Hamiltonian saddles; continuous piecewise linear differential systems; discontinuous piecewise differential systems Author: On theJaume Llibre and Claudia Valls Creator: LaTeX with hyperref Producer: pdfTeX-1.40.21 CreationDate: Fri Jun 25 06:08:56 2021 CEST ModDate: Fri Jun 25 06:12:04 2021 CEST 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: 301194 bytes Optimized: no PDF version: 1.7 name type encoding emb sub uni object ID ------------------------------------ ----------------- ---------------- --- --- --- --------- TGZVWR+URWPalladioL-Ital Type 1 Custom yes yes no 10 0 TIFBEV+URWPalladioL-Bold Type 1 Custom yes yes no 15 0 TUIUIV+URWPalladioL-Roma Type 1 Custom yes yes no 19 0 GIGFZE+CMR10 Type 1 Builtin yes yes no 23 0 NRHKIG+CMSY10 Type 1 Builtin yes yes no 27 0 QDTWCG+MSBM10 Type 1 Builtin yes yes no 31 0 GXBWHU+TimesNewRomanPS-BoldMT Type 1C WinAnsi yes yes no 39 0 CSRIGL+TimesNewRomanPS-BoldItalicMT Type 1C WinAnsi yes yes no 42 0 MCBCCI+SourceSansRoman-Regular Type 1C Custom yes yes no 64 0 MCBCCM+SourceSansRoman.613wght Type 1C Custom yes yes no 68 0 MCBCCO+SourceSansRoman-Bold Type 1C Custom yes yes no 72 0 SVZJBJ+PazoMath-Italic Type 1 Builtin yes yes no 84 0 SYFPBV+CMMI10 Type 1 Builtin yes yes no 88 0 CREATU+CMEX10 Type 1 Builtin yes yes no 95 0 KSGHMQ+PazoMath Type 1 Builtin yes yes no 99 0 Jhove (Rel. 1.22.1, 2019-04-17) Date: 2022-03-10 03:02:03 CET RepresentationInformation: symmetry_a2021v13n7p1128.pdf ReportingModule: PDF-hul, Rel. 1.12.1 (2019-04-17) LastModified: 2022-03-04 12:36:26 CET Size: 301194 Format: PDF Version: 1.7 Status: Well-Formed and valid SignatureMatches: PDF-hul MIMEtype: application/pdf PDFMetadata: Objects: 250 FreeObjects: 1 IncrementalUpdates: 0 DocumentCatalog: PageLayout: SinglePage PageMode: UseNone Outlines: Item: Title: Introduction Destination: edu.harvard.hul.ois.jhove.module.pdf.PdfSimpleObject@13c27452 Item: Title: Preliminaries Destination: edu.harvard.hul.ois.jhove.module.pdf.PdfSimpleObject@262b2c86 Children: Item: Title: Piecewise Differential Systems Destination: edu.harvard.hul.ois.jhove.module.pdf.PdfSimpleObject@371a67ec Item: Title: Linear Hamiltonian Saddles Destination: edu.harvard.hul.ois.jhove.module.pdf.PdfSimpleObject@5ed828d Item: Title: Proof of Theorem 1 Destination: edu.harvard.hul.ois.jhove.module.pdf.PdfSimpleObject@50d0686 Item: Title: Proof of Theorem 2 Destination: edu.harvard.hul.ois.jhove.module.pdf.PdfSimpleObject@7a3d45bd Item: Title: Proof of Theorem 3 Destination: edu.harvard.hul.ois.jhove.module.pdf.PdfSimpleObject@1e7c7811 Item: Title: Proof of Theorem 4 Destination: edu.harvard.hul.ois.jhove.module.pdf.PdfSimpleObject@77ec78b9 Item: Title: Discussion Destination: edu.harvard.hul.ois.jhove.module.pdf.PdfSimpleObject@1a3869f4 Item: Title: Conclusions Destination: edu.harvard.hul.ois.jhove.module.pdf.PdfSimpleObject@a38d7a3 Item: Title: References Destination: edu.harvard.hul.ois.jhove.module.pdf.PdfSimpleObject@77f99a05 Info: Title: Limit Cycles of Planar Piecewise Differential Systems with Linear Hamiltonian Saddles Author: On theJaume Llibre and Claudia Valls Subject: We provide the maximum number of limit cycles for continuous and discontinuous planar piecewise differential systems formed by linear Hamiltonian saddles and separated either by one or two parallel straight lines. We show that when these piecewise differential systems are either continuous or discontinuous and are separated by one straight line, or are continuous and are separated by two parallel straight lines, they do not have limit cycles. On the other hand, when these systems are discontinuous and separated by two parallel straight lines, we prove that the maximum number of limit cycles that they can have is one and that this maximum is reached by providing an example of such a system with one limit cycle. When the line of discontinuity of the piecewise differential system is formed by one straight line, the symmetry of the problem allows to take this straight line without loss of generality as the line x=0. Similarly, when the line of discontinuity of the piecewise differential system is formed by two parallel straight lines due to the symmetry of the problem, we can assume without loss of generality that these two straight lines are x=1. 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