Poincaré–Bendixson theorem

E156190

mathematical theorem result in dynamical systems

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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All labels observed (2)

Label	Occurrences
Poincaré–Bendixson theorem canonical	7
Poincaré–Bendixson theory	1

How this entity was disambiguated

This entity first appeared as the object of triple T1358648 — 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: Poincaré–Bendixson theorem
Context triple: [Henri Poincaré, notableWork, Poincaré–Bendixson theorem]

A. 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.
B. Brouwer fixed-point theorem
The Brouwer fixed-point theorem is a fundamental result in topology stating that any continuous function from a compact convex set (such as a closed disk) to itself has at least one fixed point.
C. local existence and uniqueness theorem
The local existence and uniqueness theorem is a fundamental result in differential equations that guarantees, under suitable conditions, a single solution passing through a given initial point, valid in some neighborhood of that point.
D. Kakutani fixed-point theorem
The Kakutani fixed-point theorem is a fundamental result in mathematical analysis and game theory that guarantees the existence of fixed points for certain set-valued (multivalued) functions, underpinning key existence proofs such as Nash equilibria.
E. Schauder fixed-point theorem
The Schauder fixed-point theorem is a fundamental result in functional analysis that guarantees the existence of fixed points for continuous compact mappings on convex closed subsets of Banach spaces, generalizing the Brouwer fixed-point theorem to infinite-dimensional settings.
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: Poincaré–Bendixson theorem
Target entity description: 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.

A. 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.
B. Brouwer fixed-point theorem
The Brouwer fixed-point theorem is a fundamental result in topology stating that any continuous function from a compact convex set (such as a closed disk) to itself has at least one fixed point.
C. local existence and uniqueness theorem
The local existence and uniqueness theorem is a fundamental result in differential equations that guarantees, under suitable conditions, a single solution passing through a given initial point, valid in some neighborhood of that point.
D. Kakutani fixed-point theorem
The Kakutani fixed-point theorem is a fundamental result in mathematical analysis and game theory that guarantees the existence of fixed points for certain set-valued (multivalued) functions, underpinning key existence proofs such as Nash equilibria.
E. Schauder fixed-point theorem
The Schauder fixed-point theorem is a fundamental result in functional analysis that guarantees the existence of fixed points for continuous compact mappings on convex closed subsets of Banach spaces, generalizing the Brouwer fixed-point theorem to infinite-dimensional settings.
F. None of above. chosen

Statements (46)

Predicate	Object
instanceOf	mathematical theorem ⓘ result in dynamical systems ⓘ
appliesTo	autonomous systems of ordinary differential equations in the plane ⓘ continuous-time dynamical systems ⓘ planar flows ⓘ two-dimensional continuous flows ⓘ
category	theorem in dynamical systems theory ⓘ theorem in ordinary differential equations ⓘ
characterizes	possible long-term behaviors of trajectories in planar systems ⓘ
concerns	alpha-limit sets ⓘ limit sets of trajectories ⓘ omega-limit sets ⓘ
conclusionType	limit set is either a fixed point, a periodic orbit, or a union of fixed points and connecting orbits ⓘ
contrastsWith	Lorenz attractor ⓘ existence of chaotic dynamics in three-dimensional flows ⓘ
dimensionRestriction	two-dimensional ⓘ
doesNotApplyTo	discrete-time dynamical systems ⓘ higher-dimensional flows ⓘ
ensures	recurrence in planar flows has restricted forms ⓘ
excludes	topologically transitive chaotic attractors in two-dimensional continuous flows ⓘ
field	dynamical systems ⓘ qualitative theory of differential equations ⓘ topology ⓘ
historicalDevelopment	originates from work of Henri Poincaré on qualitative theory of differential equations ⓘ refined and extended by Ivar Bendixson ⓘ
holdsOn	plane ⓘ two-dimensional sphere ⓘ
implies	omega-limit sets in planar flows are relatively simple ⓘ strange attractors cannot exist in two-dimensional continuous flows ⓘ
namedAfter	Henri Poincaré ⓘ Ivar Bendixson ⓘ
relatedTo	Bendixson–Dulac criterion ⓘ limit cycle ⓘ omega-limit set ⓘ planar vector field ⓘ
requires	flow defined on a two-dimensional manifold ⓘ trajectory with precompact forward orbit ⓘ
statesThat	a nonempty compact limit set of a trajectory of a C1 flow on the plane that contains no fixed point is a periodic orbit ⓘ chaotic dynamics cannot occur in two-dimensional continuous flows on the plane or sphere ⓘ nonempty compact omega-limit sets of planar flows without equilibria are periodic orbits ⓘ
typicalAssumption	vector field is continuously differentiable ⓘ
typicalFormulation	for a C1 flow on a two-dimensional manifold, a nonempty compact omega-limit set containing only finitely many equilibria is either an equilibrium, a periodic orbit, or a finite union of equilibria and connecting orbits ⓘ
usedIn	phase plane analysis ⓘ qualitative analysis of planar differential equations ⓘ stability theory of planar systems ⓘ study of limit cycles ⓘ

How these facts were elicited

Referenced by (8)

Full triples — surface form annotated when it differs from this entity's canonical label.

Henri Poincaré → notableWork → Poincaré–Bendixson theorem ⓘ

Poincaré map → relatedTo → Poincaré–Bendixson theorem ⓘ

Ivar Bendixson → notableWork → Poincaré–Bendixson theorem ⓘ

Ivar Bendixson → knownFor → Poincaré–Bendixson theorem ⓘ

Ivar Bendixson → notableConcept → Poincaré–Bendixson theorem ⓘ

Ivar Bendixson → hasNameInTheorem → Poincaré–Bendixson theorem ⓘ

Bendixson–Dulac criterion → relatedTo → Poincaré–Bendixson theorem ⓘ

Cartwright–Littlewood theory on nonlinear differential equations → relatedTo → Poincaré–Bendixson theory ⓘ

linked to: Poincaré–Bendixson theorem

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Statements (46)

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Referenced by (8)

How this entity was disambiguated

How these facts were elicited