Tightening Poincaré–Bendixson theory after counting separately the fixed points on the boundary and interior of a planar region
This paper tightens the classical Poincaré–Bendixson theory for a positively invariant, simply-connected compact set $\mathcal M$ in a continuously differentiable planar vector field by further characterizing for any point $p\in \mathcal M$, the composition of the limit sets $\omega (p)$ and $\alpha...
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| Language: | English |
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University of Szeged
2024-06-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
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| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=10796 |
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| _version_ | 1841533476749180928 |
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| author | Pouria Ramazi Ming Cao Jacquelien Scherpen |
| author_facet | Pouria Ramazi Ming Cao Jacquelien Scherpen |
| author_sort | Pouria Ramazi |
| collection | DOAJ |
| description | This paper tightens the classical Poincaré–Bendixson theory for a positively invariant, simply-connected compact set $\mathcal M$ in a continuously differentiable planar vector field by further characterizing for any point $p\in \mathcal M$, the composition of the limit sets $\omega (p)$ and $\alpha(p)$ after counting separately the fixed points on $\mathcal M$'s boundary and interior. In particular, when $\mathcal M$ contains finitely many boundary but no interior fixed points, $\omega (p)$ contains only a single fixed point, and when $\mathcal M$ may have infinitely many boundary but no interior fixed points, $\omega (p)$ can, in addition, be a continuum of fixed points. When $\mathcal M$ contains only one interior and finitely many boundary fixed points, $\omega (p)$ or $\alpha (p)$ contains exclusively a fixed point, a closed orbit or the union of the interior fixed point and homoclinic orbits joining it to itself. When $\mathcal M$ contains in general a finite number of fixed points and neither $\omega (p)$ nor $\alpha (p)$ is a closed orbit or contains just a fixed point, at least one of $\omega (p)$ and $\alpha (p)$ excludes all boundary fixed points and consists only of a number of the interior fixed points and orbits connecting them. |
| format | Article |
| id | doaj-art-3eb6eee17bf3454082df00635af55d54 |
| institution | Kabale University |
| issn | 1417-3875 |
| language | English |
| publishDate | 2024-06-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj-art-3eb6eee17bf3454082df00635af55d542025-01-15T21:24:58ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752024-06-012024291910.14232/ejqtde.2024.1.2910796Tightening Poincaré–Bendixson theory after counting separately the fixed points on the boundary and interior of a planar regionPouria Ramazi0Ming CaoJacquelien Scherpen1Brock University, St. Catharines, CanadaFaculty of Science and Engineering, University of Groningen, Groningen, NetherlandsThis paper tightens the classical Poincaré–Bendixson theory for a positively invariant, simply-connected compact set $\mathcal M$ in a continuously differentiable planar vector field by further characterizing for any point $p\in \mathcal M$, the composition of the limit sets $\omega (p)$ and $\alpha(p)$ after counting separately the fixed points on $\mathcal M$'s boundary and interior. In particular, when $\mathcal M$ contains finitely many boundary but no interior fixed points, $\omega (p)$ contains only a single fixed point, and when $\mathcal M$ may have infinitely many boundary but no interior fixed points, $\omega (p)$ can, in addition, be a continuum of fixed points. When $\mathcal M$ contains only one interior and finitely many boundary fixed points, $\omega (p)$ or $\alpha (p)$ contains exclusively a fixed point, a closed orbit or the union of the interior fixed point and homoclinic orbits joining it to itself. When $\mathcal M$ contains in general a finite number of fixed points and neither $\omega (p)$ nor $\alpha (p)$ is a closed orbit or contains just a fixed point, at least one of $\omega (p)$ and $\alpha (p)$ excludes all boundary fixed points and consists only of a number of the interior fixed points and orbits connecting them.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=10796poincaré–bendixson theoryplanar vector fieldlimit set |
| spellingShingle | Pouria Ramazi Ming Cao Jacquelien Scherpen Tightening Poincaré–Bendixson theory after counting separately the fixed points on the boundary and interior of a planar region Electronic Journal of Qualitative Theory of Differential Equations poincaré–bendixson theory planar vector field limit set |
| title | Tightening Poincaré–Bendixson theory after counting separately the fixed points on the boundary and interior of a planar region |
| title_full | Tightening Poincaré–Bendixson theory after counting separately the fixed points on the boundary and interior of a planar region |
| title_fullStr | Tightening Poincaré–Bendixson theory after counting separately the fixed points on the boundary and interior of a planar region |
| title_full_unstemmed | Tightening Poincaré–Bendixson theory after counting separately the fixed points on the boundary and interior of a planar region |
| title_short | Tightening Poincaré–Bendixson theory after counting separately the fixed points on the boundary and interior of a planar region |
| title_sort | tightening poincare bendixson theory after counting separately the fixed points on the boundary and interior of a planar region |
| topic | poincaré–bendixson theory planar vector field limit set |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=10796 |
| work_keys_str_mv | AT pouriaramazi tighteningpoincarebendixsontheoryaftercountingseparatelythefixedpointsontheboundaryandinteriorofaplanarregion AT mingcao tighteningpoincarebendixsontheoryaftercountingseparatelythefixedpointsontheboundaryandinteriorofaplanarregion AT jacquelienscherpen tighteningpoincarebendixsontheoryaftercountingseparatelythefixedpointsontheboundaryandinteriorofaplanarregion |