Bendixson–Dulac criterion
E620667
The Bendixson–Dulac criterion is a result in the qualitative theory of planar dynamical systems that provides conditions under which a system has no periodic orbits in a given region.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Bendixson–Dulac criterion canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6801241 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Bendixson–Dulac criterion Context triple: [Poincaré–Bendixson theorem, relatedTo, Bendixson–Dulac criterion]
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A.
Poincaré–Bendixson theorem
The Poincaré–Bendixson theorem is a fundamental result in the qualitative theory of dynamical systems that characterizes the possible long-term behaviors of trajectories in two-dimensional continuous flows, ruling out chaotic dynamics in the plane.
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B.
Poincaré–Birkhoff fixed-point theorem
The Poincaré–Birkhoff fixed-point theorem is a fundamental result in dynamical systems and topology that guarantees the existence of at least two fixed points for certain area-preserving twist maps of an annulus.
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C.
Cauchy–Kovalevskaya theorem
The Cauchy–Kovalevskaya theorem is a fundamental result in partial differential equations that guarantees the existence and uniqueness of analytic solutions to certain initial value problems under appropriate analyticity conditions.
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D.
Peano existence theorem
The Peano existence theorem is a fundamental result in the theory of ordinary differential equations that guarantees the existence (but not necessarily uniqueness) of solutions under mild continuity conditions on the right-hand side.
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E.
Routh–Hurwitz stability criterion
The Routh–Hurwitz stability criterion is a mathematical test in control theory that determines whether all roots of a system’s characteristic polynomial lie in the left half of the complex plane, ensuring system stability without explicitly computing the roots.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Bendixson–Dulac criterion Target entity description: The Bendixson–Dulac criterion is a result in the qualitative theory of planar dynamical systems that provides conditions under which a system has no periodic orbits in a given region.
-
A.
Poincaré–Bendixson theorem
The Poincaré–Bendixson theorem is a fundamental result in the qualitative theory of dynamical systems that characterizes the possible long-term behaviors of trajectories in two-dimensional continuous flows, ruling out chaotic dynamics in the plane.
-
B.
Poincaré–Birkhoff fixed-point theorem
The Poincaré–Birkhoff fixed-point theorem is a fundamental result in dynamical systems and topology that guarantees the existence of at least two fixed points for certain area-preserving twist maps of an annulus.
-
C.
Cauchy–Kovalevskaya theorem
The Cauchy–Kovalevskaya theorem is a fundamental result in partial differential equations that guarantees the existence and uniqueness of analytic solutions to certain initial value problems under appropriate analyticity conditions.
-
D.
Peano existence theorem
The Peano existence theorem is a fundamental result in the theory of ordinary differential equations that guarantees the existence (but not necessarily uniqueness) of solutions under mild continuity conditions on the right-hand side.
-
E.
Routh–Hurwitz stability criterion
The Routh–Hurwitz stability criterion is a mathematical test in control theory that determines whether all roots of a system’s characteristic polynomial lie in the left half of the complex plane, ensuring system stability without explicitly computing the roots.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in dynamical systems ⓘ |
| appliesTo |
planar autonomous systems
ⓘ
systems of the form x' = P(x,y), y' = Q(x,y) ⓘ two-dimensional differential equations ⓘ |
| assumes |
a continuously differentiable vector field
ⓘ
a simply connected region in the plane ⓘ |
| category | nonexistence theorem ⓘ |
| conclusion | no periodic orbit lies entirely in the interior of the region ⓘ |
| conditionType | sufficient condition for nonexistence of periodic orbits ⓘ |
| contrastWith | results that guarantee existence of periodic orbits ⓘ |
| field |
planar dynamical systems
ⓘ
qualitative theory of dynamical systems ⓘ |
| generalizationOf | Bendixson nonexistence criterion NERFINISHED ⓘ |
| generalizes | Bendixson criterion NERFINISHED ⓘ |
| historicalPeriod | early 20th century ⓘ |
| implies |
absence of closed trajectories in the region
ⓘ
absence of limit cycles in the region ⓘ |
| languageOfFormulation |
real analysis
ⓘ
vector calculus ⓘ |
| mainStatement | under certain conditions a planar system has no periodic orbits in a given region ⓘ |
| mathematicalDomain |
ordinary differential equations
ⓘ
phase plane analysis ⓘ |
| namedAfter |
Henri Dulac
NERFINISHED
ⓘ
Ivar Bendixson NERFINISHED ⓘ |
| not | necessary condition for nonexistence of periodic orbits ⓘ |
| objectOfStudy | closed trajectories of planar vector fields ⓘ |
| proofTechnique |
Green's theorem
NERFINISHED
ⓘ
integral of divergence over a region ⓘ |
| purpose |
to rule out the existence of periodic orbits
ⓘ
to study global phase portrait of planar systems ⓘ |
| relatedTo |
Hilbert's sixteenth problem
NERFINISHED
ⓘ
Poincaré–Bendixson theorem NERFINISHED ⓘ limit cycle theory ⓘ |
| requires |
divergence of the modified vector field has constant sign except possibly on a set of measure zero
ⓘ
existence of a C1 Dulac function on the region ⓘ nonvanishing Dulac function on the region ⓘ |
| typicalAssumption |
region considered is simply connected
ⓘ
vector field is continuously differentiable on an open subset of R2 ⓘ |
| usedFor |
excluding periodic behavior in planar models
ⓘ
proving global stability of equilibria ⓘ |
| usedIn |
chemical reaction dynamics
ⓘ
control theory planar systems ⓘ ecological dynamical systems ⓘ mathematical biology models ⓘ |
| usesConcept |
Dulac function
NERFINISHED
ⓘ
divergence of a vector field ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Bendixson–Dulac criterion Description of subject: The Bendixson–Dulac criterion is a result in the qualitative theory of planar dynamical systems that provides conditions under which a system has no periodic orbits in a given region.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.