Poincaré map

E156191

mathematical concept tool in dynamical systems theory

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.

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All labels observed (3)

Label	Occurrences
Poincaré map canonical	1
Poincaré return map	1
Poincaré section	1

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 ⓘ

Referenced by (3)

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

Henri Poincaré → notableWork → Poincaré map ⓘ

Poincaré map → alsoKnownAs → Poincaré map ⓘ

this entity surface form: Poincaré section

Poincaré map → alsoKnownAs → Poincaré map ⓘ

this entity surface form: Poincaré return map