Poincaré–Bendixson theorem
E156190
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Poincaré–Bendixson theorem canonical | 2 |
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]
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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.
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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.
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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
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Instruction
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Input
Subject: Poincaré–Bendixson theorem Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.