Poincaré map
E156191
The Poincaré map is a mathematical tool in dynamical systems theory that reduces continuous-time dynamics to a discrete map by tracking intersections of trajectories with a lower-dimensional surface.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Poincaré map canonical | 1 |
| Poincaré return map | 1 |
| Poincaré section | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1358650 — 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 map Context triple: [Henri Poincaré, notableWork, Poincaré map]
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A.
Gauss map
The Gauss map is a differential geometry concept that assigns to each point on a surface the corresponding point on the unit sphere determined by the surface’s normal vector at that point.
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B.
Mapam
Mapam was a left-wing socialist Zionist political party in Israel that played a significant role in the early decades of the state, particularly within the kibbutz movement and peace-oriented politics.
-
C.
Ulam spiral
The Ulam spiral is a graphical arrangement of the positive integers in a spiral pattern that reveals striking diagonal alignments of prime numbers, suggesting unexpected structure in their distribution.
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D.
Carathéodory–Jacobi–Lie theorem
The Carathéodory–Jacobi–Lie theorem is a fundamental result in symplectic geometry and Hamiltonian mechanics that provides canonical local coordinates adapted to a given set of commuting functions.
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E.
#MAPA
#MAPA is a hashtag used within the Fridays for Future movement to highlight and organize climate activism by communities in the Most Affected People and Areas.
- 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 map Target entity description: The Poincaré map is a mathematical tool in dynamical systems theory that reduces continuous-time dynamics to a discrete map by tracking intersections of trajectories with a lower-dimensional surface.
-
A.
Gauss map
The Gauss map is a differential geometry concept that assigns to each point on a surface the corresponding point on the unit sphere determined by the surface’s normal vector at that point.
-
B.
Mapam
Mapam was a left-wing socialist Zionist political party in Israel that played a significant role in the early decades of the state, particularly within the kibbutz movement and peace-oriented politics.
-
C.
Ulam spiral
The Ulam spiral is a graphical arrangement of the positive integers in a spiral pattern that reveals striking diagonal alignments of prime numbers, suggesting unexpected structure in their distribution.
-
D.
Carathéodory–Jacobi–Lie theorem
The Carathéodory–Jacobi–Lie theorem is a fundamental result in symplectic geometry and Hamiltonian mechanics that provides canonical local coordinates adapted to a given set of commuting functions.
-
E.
#MAPA
#MAPA is a hashtag used within the Fridays for Future movement to highlight and organize climate activism by communities in the Most Affected People and Areas.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical concept
ⓘ
tool in dynamical systems theory ⓘ |
| alsoKnownAs |
Poincaré map
ⓘ
surface form:
Poincaré return map
Poincaré map ⓘ
surface form:
Poincaré section
first return map ⓘ |
| appliedIn |
bifurcation analysis
ⓘ
celestial mechanics ⓘ chaos detection ⓘ control theory ⓘ mechanical systems with impacts ⓘ nonlinear oscillations ⓘ study of limit cycles ⓘ |
| assumes | transversality of the section to the flow ⓘ |
| basedOn | intersections of trajectories with a surface ⓘ |
| canBe |
defined locally near a periodic orbit
ⓘ
iterated to study long-term behavior ⓘ |
| characteristic |
captures recurrence properties of trajectories
ⓘ
dimension reduction technique ⓘ discrete-time representation of a continuous flow ⓘ often defined on a hypersurface of codimension one ⓘ |
| codomain | lower-dimensional surface ⓘ |
| domain | continuous-time dynamical system ⓘ |
| field |
differential equations
ⓘ
dynamical systems ⓘ mathematical physics ⓘ |
| helpsIdentify |
bifurcations of periodic solutions
ⓘ
fixed points corresponding to periodic orbits ⓘ invariant sets on the section ⓘ |
| historicalContext | introduced in late 19th century ⓘ |
| input | point on the Poincaré section ⓘ |
| mathematicalNature | discrete dynamical system ⓘ |
| namedAfter | Henri Poincaré ⓘ |
| output | next intersection of the trajectory with the section ⓘ |
| purpose |
analyze periodic orbits
ⓘ
detect stability of periodic solutions ⓘ reduce continuous-time dynamics to a discrete map ⓘ simplify phase space analysis ⓘ study qualitative behavior of dynamical systems ⓘ |
| relatedTo |
Poincaré–Bendixson theorem
ⓘ
flow of a vector field ⓘ phase space ⓘ return map ⓘ stroboscopic map ⓘ |
| typicalSection | surface of codimension one in phase space ⓘ |
| usedFor |
computation of Floquet multipliers
ⓘ
numerical investigation of dynamical systems ⓘ stability analysis of periodic orbits ⓘ visualization of chaotic attractors ⓘ |
| uses | transversal section to the flow ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Poincaré map Description of subject: The Poincaré map is a mathematical tool in dynamical systems theory that reduces continuous-time dynamics to a discrete map by tracking intersections of trajectories with a lower-dimensional surface.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Poincaré section
this entity surface form:
Poincaré return map