a656998941eba584daad88aad4a10472 symmetry_a2021v13n9p1736.pdf 4f849609510761d49724864fd905530398f3017f symmetry_a2021v13n9p1736.pdf 0f15075607e9f0197e1122e946270be0eea97d2a09a3138986b34a3f280b721c symmetry_a2021v13n9p1736.pdf Title: Integrability and Limit Cycles via First Integrals Subject: In many problems appearing in applied mathematics in the nonlinear ordinary differential systems, as in physics, chemist, economics, etc., if we have a differential system on a manifold of dimension, two of them having a first integral, then its phase portrait is completely determined. While the existence of first integrals for differential systems on manifolds of a dimension higher than two allows to reduce the dimension of the space in as many dimensions as independent first integrals we have. Hence, to know first integrals is important, but the following question appears: Given a differential system, how to know if it has a first integral? The symmetries of many differential systems force the existence of first integrals. This paper has two main objectives. First, we study how to compute first integrals for polynomial differential systems using the so-called Darboux theory of integrability. Furthermore, second, we show how to use the existence of first integrals for finding limit cycles in piecewise differential systems. Keywords: limit cycles; Darboux theory of integrability; first integrals Author: Jaume Llibre Creator: LaTeX with hyperref Producer: pdfTeX-1.40.21 CreationDate: Wed Sep 22 10:51:06 2021 CEST ModDate: Thu Sep 23 07:26:02 2021 CEST Tagged: no UserProperties: no Suspects: no Form: none JavaScript: no Pages: 21 Encrypted: no Page size: 595.276 x 841.89 pts (A4) Page rot: 0 File size: 430495 bytes Optimized: no PDF version: 1.7 name type encoding emb sub uni object ID ------------------------------------ ----------------- ---------------- --- --- --- --------- VBFFDA+URWPalladioL-Ital Type 1 Custom yes yes no 10 0 XBJLVY+URWPalladioL-Bold Type 1 Custom yes yes no 15 0 VSAKWV+URWPalladioL-Roma Type 1 Custom yes yes no 19 0 SOSTRQ+CMR10 Type 1 Builtin yes yes no 23 0 WBWAJW+PazoMath-Italic Type 1 Builtin yes yes no 27 0 UCPGSO+CMSY10 Type 1 Builtin yes yes no 31 0 VAXAEC+MSBM10 Type 1 Builtin yes yes no 35 0 GXBWHU+TimesNewRomanPS-BoldMT Type 1C WinAnsi yes yes no 43 0 CSRIGL+TimesNewRomanPS-BoldItalicMT Type 1C WinAnsi yes yes no 46 0 MCBCCI+SourceSansRoman-Regular Type 1C Custom yes yes no 68 0 MCBCCM+SourceSansRoman.613wght Type 1C Custom yes yes no 72 0 MCBCCO+SourceSansRoman-Bold Type 1C Custom yes yes no 76 0 WTQPOO+CMEX10 Type 1 Builtin yes yes no 88 0 YYMMJX+PazoMath Type 1 Builtin yes yes no 98 0 SYFPBV+CMMI10 Type 1 Builtin yes yes no 105 0 UFZCWT+f-1-0 Type 1C WinAnsi yes yes no 152 0 VHGNDT+f-0-0 Type 1C WinAnsi yes yes no 155 0 ITPLBX+ArialMT TrueType WinAnsi yes yes yes 167 0 MQHRED+MathematicaSans TrueType WinAnsi yes yes yes 171 0 Jhove (Rel. 1.22.1, 2019-04-17) Date: 2022-03-10 03:04:59 CET RepresentationInformation: symmetry_a2021v13n9p1736.pdf ReportingModule: PDF-hul, Rel. 1.12.1 (2019-04-17) LastModified: 2022-02-04 12:47:15 CET Size: 430495 Format: PDF Version: 1.7 Status: Well-Formed and valid SignatureMatches: PDF-hul MIMEtype: application/pdf PDFMetadata: Objects: 377 FreeObjects: 1 IncrementalUpdates: 0 DocumentCatalog: PageLayout: SinglePage PageMode: UseNone Outlines: Item: Title: Introduction to the Darboux Theory of Integrability Destination: edu.harvard.hul.ois.jhove.module.pdf.PdfSimpleObject@13c27452 Children: Item: Title: Polynomial Differential Systems Destination: edu.harvard.hul.ois.jhove.module.pdf.PdfSimpleObject@262b2c86 Item: Title: First Integrals Destination: edu.harvard.hul.ois.jhove.module.pdf.PdfSimpleObject@371a67ec Item: Title: Integrating Factors Destination: edu.harvard.hul.ois.jhove.module.pdf.PdfSimpleObject@5ed828d Item: Title: Invariant Algebraic Curves Destination: edu.harvard.hul.ois.jhove.module.pdf.PdfSimpleObject@50d0686 Item: Title: Exponential Factors Destination: edu.harvard.hul.ois.jhove.module.pdf.PdfSimpleObject@7a3d45bd Item: Title: The Method of Darboux Destination: edu.harvard.hul.ois.jhove.module.pdf.PdfSimpleObject@1e7c7811 Item: Title: Limit Cycles in Piecewise Differential Systems via First Integrals Destination: edu.harvard.hul.ois.jhove.module.pdf.PdfSimpleObject@77ec78b9 Children: Item: Title: Discontinuous Piecewise Differential Systems Destination: edu.harvard.hul.ois.jhove.module.pdf.PdfSimpleObject@1a3869f4 Item: Title: Limit Cycles of a Piecewise Differential System Formed by a Linear Differential System and a Quadratic Polynomial Differential System Separated by the Straight Line x=0 Destination: edu.harvard.hul.ois.jhove.module.pdf.PdfSimpleObject@a38d7a3 Item: Title: Limit Cycles of Piecewise Differential Systems Formed by Three Linear Centers Destination: edu.harvard.hul.ois.jhove.module.pdf.PdfSimpleObject@77f99a05 Item: Title: Periodic Orbits of a Relay System in R3 Destination: edu.harvard.hul.ois.jhove.module.pdf.PdfSimpleObject@63440df3 Item: Title: Limit Cycles of a Class of Piecewise Differential Systems Separated by a Parabola Destination: edu.harvard.hul.ois.jhove.module.pdf.PdfSimpleObject@3aeaafa6 Item: Title: Piecewise Differential System with a Non-Regular Discontinuity Line Destination: edu.harvard.hul.ois.jhove.module.pdf.PdfSimpleObject@76a3e297 Item: Title: Discussion Destination: edu.harvard.hul.ois.jhove.module.pdf.PdfSimpleObject@4d3167f4 Item: Title: Conclusions Destination: edu.harvard.hul.ois.jhove.module.pdf.PdfSimpleObject@ed9d034 Item: Title: References Destination: edu.harvard.hul.ois.jhove.module.pdf.PdfSimpleObject@6121c9d6 Info: Title: Integrability and Limit Cycles via First Integrals Author: Jaume Llibre Subject: In many problems appearing in applied mathematics in the nonlinear ordinary differential systems, as in physics, chemist, economics, etc., if we have a differential system on a manifold of dimension, two of them having a first integral, then its phase portrait is completely determined. While the existence of first integrals for differential systems on manifolds of a dimension higher than two allows to reduce the dimension of the space in as many dimensions as independent first integrals we have. Hence, to know first integrals is important, but the following question appears: Given a differential system, how to know if it has a first integral? The symmetries of many differential systems force the existence of first integrals. This paper has two main objectives. First, we study how to compute first integrals for polynomial differential systems using the so-called Darboux theory of integrability. Furthermore, second, we show how to use the existence of first integrals for finding limit cycles in piecewise differential systems. 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