Tightening Poincaré–Bendixson theory after counting separately the fixed points on the boundary and interior of a planar region
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Abstract
This paper tightens the classical Poincaré–Bendixson theory for a positively invariant, simply-connected compact set <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow class="MJX-TeXAtom-ORD"> <a:mi class="MJX-tex-caligraphic" mathvariant="script">M</a:mi> </a:mrow> </a:math> in a continuously differentiable planar vector field by further characterizing for any point <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mi>p</e:mi> <e:mo>∈</e:mo> <e:mrow class="MJX-TeXAtom-ORD"> <e:mi class="MJX-tex-caligraphic" mathvariant="script">M</e:mi> </e:mrow> </e:math> , the composition of the limit sets <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"> <i:mi>ω</i:mi> <i:mo stretchy="false">(</i:mo> <i:mi>p</i:mi> <i:mo stretchy="false">)</i:mo> </i:math> and <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"> <l:mi>α</l:mi> <l:mo stretchy="false">(</l:mo> <l:mi>p</l:mi> <l:mo stretchy="false">)</l:mo> </l:math> after counting separately the fixed points on <o:math xmlns:o="http://www.w3.org/1998/Math/MathML"> <o:mrow class="MJX-TeXAtom-ORD"> <o:mi class="MJX-tex-caligraphic" mathvariant="script">M</o:mi> </o:mrow> </o:math> 's boundary and interior. In particular, when <s:math xmlns:s="http://www.w3.org/1998/Math/MathML"> <s:mrow class="MJX-TeXAtom-ORD"> <s:mi class="MJX-tex-caligraphic" mathvariant="script">M</s:mi> </s:mrow> </s:math> contains finitely many boundary but no interior fixed points, <w:math xmlns:w="http://www.w3.org/1998/Math/MathML"> <w:mi>ω</w:mi> <w:mo stretchy="false">(</w:mo> <w:mi>p</w:mi> <w:mo stretchy="false">)</w:mo> </w:math> contains only a single fixed point, and when <z:math xmlns:z="http://www.w3.org/1998/Math/MathML"> <z:mrow class="MJX-TeXAtom-ORD"> <z:mi class="MJX-tex-caligraphic" mathvariant="script">M</z:mi> </z:mrow> </z:math> may have infinitely many boundary but no interior fixed points, <db:math xmlns:db="http://www.w3.org/1998/Math/MathML"> <db:mi>ω</db:mi> <db:mo stretchy="false">(</db:mo> <db:mi>p</db:mi> <db:mo stretchy="false">)</db:mo> </db:math> can, in addition, be a continuum of fixed points. When <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML"> <gb:mrow class="MJX-TeXAtom-ORD"> <gb:mi class="MJX-tex-caligraphic" mathvariant="script">M</gb:mi> </gb:mrow> </gb:math> contains only one interior and finitely many boundary fixed points, <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML"> <kb:mi>ω</kb:mi> <kb:mo stretchy="false">(</kb:mo> <kb:mi>p</kb:mi> <kb:mo stretchy="false">)</kb:mo> </kb:math> or <nb:math xmlns:nb="http://www.w3.org/1998/Math/MathML"> <nb:mi>α</nb:mi> <nb:mo stretchy="false">(</nb:mo> <nb:mi>p</nb:mi> <nb:mo stretchy="false">)</nb:mo> </nb:math> contains exclusively a fixed point, a closed orbit or the union of the interior fixed point and homoclinic orbits joining it to itself. When <qb:math xmlns:qb="http://www.w3.org/1998/Math/MathML"> <qb:mrow class="MJX-TeXAtom-ORD"> <qb:mi class="MJX-tex-caligraphic" mathvariant="script">M</qb:mi> </qb:mrow> </qb:math> contains in general a finite number of fixed points and neither <ub:math xmlns:ub="http://www.w3.org/1998/Math/MathML"> <ub:mi>ω</ub:mi> <ub:mo stretchy="false">(</ub:mo> <ub:mi>p</ub:mi> <ub:mo stretchy="false">)</ub:mo> </ub:math> nor <xb:math xmlns:xb="http://www.w3.org/1998/Math/MathML"> <xb:mi>α</xb:mi> <xb:mo stretchy="false">(</xb:mo> <xb:mi>p</xb:mi> <xb:mo stretchy="false">)</xb:mo> </xb:math> is a closed orbit or contains just a fixed point, at least one of <ac:math xmlns:ac="http://www.w3.org/1998/Math/MathML"> <ac:mi>ω</ac:mi> <ac:mo stretchy="false">(</ac:mo> <ac:mi>p</ac:mi> <ac:mo stretchy="false">)</ac:mo> </ac:math> and <dc:math xmlns:dc="http://www.w3.org/1998/Math/MathML"> <dc:mi>α</dc:mi> <dc:mo stretchy="false">(</dc:mo> <dc:mi>p</dc:mi> <dc:mo stretchy="false">)</dc:mo> </dc:math> excludes all boundary fixed points and consists only of a number of the interior fixed points and orbits connecting them.
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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