25fe473c2fb1f4f4ae500824da263091 appliedmath-05-00068-v2.pdf fa2159342514720bab65a78d01d12f64dc01b966 appliedmath-05-00068-v2.pdf f48256c9fd832386de54509b6d8acb339151aa196a50545ce362f0f077f42540 appliedmath-05-00068-v2.pdf Title: New Exploration of Phase Portrait Classification of Quadratic Polynomial Differential Systems Based on Invariant Theory Subject: After linear differential systems in the plane, the easiest systems are quadratic polynomial differential systems in the plane. Due to their nonlinearity and their many applications, these systems have been studied by many authors. Such quadratic polynomial differential systems have been divided into ten families. Here, for two of these families, we classify all topologically distinct phase portraits in the Poincaré disc. These two families have already been studied previously, but several mistakes made there are repaired here thanks to the use of a more powerful technique. This new technique uses the invariant theory developed by the Sibirskii School, applied to differential systems, which allows to determine all the algebraic bifurcations in a relatively easy way. Even though the goal of obtaining all the phase portraits of quadratic systems for each of the ten families is not achievable using only this method, the coordination of different approaches may help us reach this goal. Keywords: quadratic vector field; quadratic system; phase portrait Author: Joan Carles Artés, Laurent Cairó and Jaume Llibre Creator: LaTeX with hyperref Producer: pdfTeX-1.40.25 CreationDate: Fri Jun 13 06:59:24 2025 CEST ModDate: Fri Jun 13 07:02:01 2025 CEST Custom Metadata: no Metadata Stream: no Tagged: no UserProperties: no Suspects: no Form: none JavaScript: no Pages: 22 Encrypted: no Page size: 595.276 x 841.89 pts (A4) Page rot: 0 File size: 577262 bytes Optimized: no PDF version: 1.7 name type encoding emb sub uni object ID ------------------------------------ ----------------- ---------------- --- --- --- --------- ZNNTGL+URWPalladioL-Roma Type 1 Custom yes yes yes 10 0 CGTXMU+URWPalladioL-Bold Type 1 Custom yes yes yes 16 0 TXIYVE+URWPalladioL-Ital Type 1 Custom yes yes yes 21 0 SOSTRQ+CMR10 Type 1 Builtin yes yes yes 26 0 VUABVQ+CMMI10 Type 1 Builtin yes yes yes 31 0 CKXAJB+MSBM10 Type 1 Builtin yes yes yes 36 0 OGPNJE+CMSY10 Type 1 Builtin yes yes yes 62 0 VEFTPH+ArialMT Type 1C WinAnsi yes yes no 75 0 OAQYEM+ArialMT Type 1C WinAnsi yes yes no 86 0 HLYWJI+CMEX10 Type 1 Builtin yes yes yes 101 0 YBQGLV+PazoMath-Italic Type 1 Builtin yes yes yes 106 0 TNJENU+PazoMath Type 1 Builtin yes yes yes 114 0 ZURFJC+ArialMT Type 1C Custom yes yes yes 125 0 FMSULL+ArialMT Type 1C Custom yes yes yes 136 0 CCSIMC+ArialMT Type 1C Custom yes yes no 148 0 EGDVVW+ArialMT Type 1C WinAnsi yes yes no 155 0 HYNTTS+ArialMT Type 1C Custom yes yes no 164 0 QJDLON+ArialMT Type 1C Custom yes yes yes 171 0 FDWLIF+ArialMT Type 1C WinAnsi yes yes no 182 0 BSHKLI+CMBSY10 Type 1 Builtin yes yes yes 194 0 OBRUTN+CMBX10 Type 1 Builtin yes yes yes 199 0 Jhove (Rel. 1.28.0, 2023-05-18) Date: 2025-07-09 02:10:06 CEST RepresentationInformation: appliedmath-05-00068-v2.pdf ReportingModule: PDF-hul, Rel. 1.12.4 (2023-03-16) LastModified: 2025-07-08 13:22:36 CEST Size: 577262 Format: PDF Version: 1.7 Status: Well-Formed and valid SignatureMatches: PDF-hul MIMEtype: application/pdf PDFMetadata: Objects: 378 FreeObjects: 1 IncrementalUpdates: 0 DocumentCatalog: PageLayout: SinglePage PageMode: UseNone Outlines: Item: Title: Introduction and Statement of the Main Results Destination: section.1 Item: Title: Preliminary Efinitions Destination: section.2 Children: Item: Title: Equilibrium Points Destination: subsection.2.1 Item: Title: Reducing the Number of Parameters of Systems VII and VIII Destination: subsection.2.2 Item: Title: Invariants Destination: subsection.2.3 Children: Item: Title: Algebraic Bifurcation Surfaces Destination: subsubsection.2.3.1 Item: Title: Non-Algebraic Bifurcation Hypersurfaces Destination: subsubsection.2.3.2 Item: Title: Differences from Previous Works Using the Same Technique Destination: subsubsection.2.3.3 Item: Title: Phase Portraits Destination: section.3 Children: Item: Title: Phase Portraits of Systems VII(A) Destination: subsection.3.1 Item: Title: Phase Portraits of Systems VII(B) Destination: subsection.3.2 Item: Title: Phase Portraits of Systems VII(C) and VII(D) Destination: subsection.3.3 Item: Title: Phase Portraits of Systems VIII(A) Destination: subsection.3.4 Item: Title: Phase Portraits of Systems VIII(B) Destination: subsection.3.5 Item: Title: Phase Portraits of Systems VIII(C) Destination: subsection.3.6 Item: Title: Global Geometrical Properties of Family VII Destination: section.4 Item: Title: Global Geometrical Properties of Family VIII Destination: section.5 Item: Title: Conclusions and Comments Destination: section.6 Item: Title: Appendix A Destination: appendix.A. Item: Title: References Destination: appendix.B. Info: Title: New Exploration of Phase Portrait Classification of Quadratic Polynomial Differential Systems Based on Invariant Theory Author: Joan Carles Artés, Laurent Cairó and Jaume Llibre Subject: After linear differential systems in the plane, the easiest systems are quadratic polynomial differential systems in the plane. Due to their nonlinearity and their many applications, these systems have been studied by many authors. Such quadratic polynomial differential systems have been divided into ten families. Here, for two of these families, we classify all topologically distinct phase portraits in the Poincaré disc. These two families have already been studied previously, but several mistakes made there are repaired here thanks to the use of a more powerful technique. This new technique uses the invariant theory developed by the Sibirskii School, applied to differential systems, which allows to determine all the algebraic bifurcations in a relatively easy way. Even though the goal of obtaining all the phase portraits of quadratic systems for each of the ten families is not achievable using only this method, the coordination of different approaches may help us reach this goal. 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