Lyapunov exponents
E181623
Lyapunov exponents are quantitative measures in dynamical systems theory that characterize the rates at which nearby trajectories diverge or converge, indicating the presence and strength of chaos.
All labels observed (5)
| Label | Occurrences |
|---|---|
| Lyapunov exponent | 7 |
| Lyapunov spectrum | 3 |
| Lyapunov exponents canonical | 2 |
| Lyapunov characteristic exponent | 1 |
| Oseledec multiplicative ergodic theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1597809 — 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: Lyapunov exponents Context triple: [Aleksandr Lyapunov, notableWork, Lyapunov exponents]
-
A.
Onsager–Machlup function
The Onsager–Machlup function is a functional in stochastic process theory that characterizes the most probable paths of fluctuating systems, playing a key role in nonequilibrium statistical mechanics and large deviation theory.
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B.
Kakutani equivalence in ergodic theory
Kakutani equivalence in ergodic theory is a notion of equivalence between measure-preserving dynamical systems based on the isomorphism of their induced transformations on subsets of positive measure.
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C.
Poincaré map
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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D.
Poincaré recurrence theorem
The Poincaré recurrence theorem is a fundamental result in dynamical systems and ergodic theory stating that certain systems will, after a sufficiently long but finite time, return arbitrarily close to their initial state.
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E.
Rényi entropy
Rényi entropy is a generalized measure of information and uncertainty that extends Shannon entropy by introducing a tunable order parameter to emphasize different aspects of a probability distribution.
- 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: Lyapunov exponents Target entity description: Lyapunov exponents are quantitative measures in dynamical systems theory that characterize the rates at which nearby trajectories diverge or converge, indicating the presence and strength of chaos.
-
A.
Onsager–Machlup function
The Onsager–Machlup function is a functional in stochastic process theory that characterizes the most probable paths of fluctuating systems, playing a key role in nonequilibrium statistical mechanics and large deviation theory.
-
B.
Kakutani equivalence in ergodic theory
Kakutani equivalence in ergodic theory is a notion of equivalence between measure-preserving dynamical systems based on the isomorphism of their induced transformations on subsets of positive measure.
-
C.
Poincaré map
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.
-
D.
Poincaré recurrence theorem
The Poincaré recurrence theorem is a fundamental result in dynamical systems and ergodic theory stating that certain systems will, after a sufficiently long but finite time, return arbitrarily close to their initial state.
-
E.
Rényi entropy
Rényi entropy is a generalized measure of information and uncertainty that extends Shannon entropy by introducing a tunable order parameter to emphasize different aspects of a probability distribution.
- F. None of above. chosen
Statements (56)
| Predicate | Object |
|---|---|
| instanceOf |
invariant of dynamical systems
ⓘ
mathematical concept ⓘ measure of chaos ⓘ |
| appliedIn |
celestial mechanics
ⓘ
climate dynamics ⓘ economics and finance ⓘ engineering control systems ⓘ mechanical systems ⓘ neuroscience ⓘ weather prediction ⓘ |
| appliesTo |
continuous-time dynamical systems
ⓘ
discrete-time dynamical systems ⓘ flows ⓘ maps ⓘ |
| characterize |
average exponential rates of separation of nearby trajectories
ⓘ
sensitivity to initial conditions ⓘ stability properties of orbits ⓘ |
| computedBy |
Benettin algorithm
ⓘ
QR decomposition methods ⓘ linearization of the flow or map ⓘ time series reconstruction methods ⓘ |
| definedAs | asymptotic time average of logarithm of growth rate of tangent vectors ⓘ |
| definedFor | each principal direction in tangent space ⓘ |
| field |
chaos theory
ⓘ
dynamical systems theory ⓘ ergodic theory ⓘ nonlinear dynamics ⓘ |
| hasType |
Lyapunov exponents
self-linksurface differs
ⓘ
surface form:
Lyapunov spectrum
finite-time Lyapunov exponent ⓘ global Lyapunov exponent ⓘ local Lyapunov exponent ⓘ maximal Lyapunov exponent ⓘ |
| indicate |
presence of chaos when positive
ⓘ
rate of convergence of nearby trajectories ⓘ rate of divergence of nearby trajectories ⓘ |
| namedAfter | Aleksandr Lyapunov ⓘ |
| property |
can form a spectrum of values for multidimensional systems
ⓘ
largest exponent often dominates long-term behavior ⓘ negative exponents correspond to contracting directions ⓘ positive exponents correspond to expanding directions ⓘ zero exponent typically corresponds to motion along the trajectory ⓘ |
| relatedTo |
Kolmogorov–Sinai entropy
ⓘ
Lyapunov dimension ⓘ Lyapunov stability ⓘ Oseledets theorem ⓘ Pesin theory ⓘ bifurcation theory ⓘ chaotic attractor ⓘ ergodic measures ⓘ sensitive dependence on initial conditions ⓘ |
| usedFor |
classifying stability of fixed points
ⓘ
classifying stability of periodic orbits ⓘ detecting strange attractors ⓘ estimating predictability horizons ⓘ quantifying chaos in dynamical systems ⓘ testing for deterministic chaos in time series ⓘ |
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: Lyapunov exponents Description of subject: Lyapunov exponents are quantitative measures in dynamical systems theory that characterize the rates at which nearby trajectories diverge or converge, indicating the presence and strength of chaos.
Referenced by (14)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Lyapunov exponent
this entity surface form:
Lyapunov exponent
this entity surface form:
Lyapunov exponent
this entity surface form:
Lyapunov spectrum
this entity surface form:
Lyapunov exponent
this entity surface form:
Lyapunov spectrum
this entity surface form:
Lyapunov exponent
this entity surface form:
Lyapunov spectrum
this entity surface form:
Lyapunov characteristic exponent
this entity surface form:
Oseledec multiplicative ergodic theorem
this entity surface form:
Lyapunov exponent
this entity surface form:
Lyapunov exponent