Bendixson–Dulac criterion

E620667

mathematical theorem result in dynamical systems

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.

Try in SPARQL Jump to: Surface forms Disambiguation Statements Elicitation Referenced by

All labels observed (6)

Label	Occurrences
Bendixson–Dulac theorem	2
Bendixson’s criterion	2
Bendixson criterion	1
Bendixson nonexistence criterion	1
Bendixson–Dulac criterion canonical	1
Dulac function	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]

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
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 ⓘ linked to: Bendixson–Dulac criterion
generalizes	Bendixson criterion ⓘ linked to: Bendixson–Dulac criterion
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 ⓘ Ivar Bendixson ⓘ
not	necessary condition for nonexistence of periodic orbits ⓘ
objectOfStudy	closed trajectories of planar vector fields ⓘ
proofTechnique	Green's theorem ⓘ 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 ⓘ linked to: Hilbert’s sixteenth problem Poincaré–Bendixson theorem ⓘ 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 ⓘ linked to: Bendixson–Dulac criterion divergence of a vector field ⓘ

How these facts were elicited