Lyapunov dimension
E695938
The Lyapunov dimension is a fractal dimension used in dynamical systems theory to quantify the effective number of degrees of freedom of a chaotic attractor based on its Lyapunov exponents.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Lyapunov dimension canonical | 2 |
| Kaplan–Yorke Lyapunov dimension | 1 |
| Kaplan–Yorke conjecture | 1 |
| Kaplan–Yorke dimension | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7833156 — 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 dimension Context triple: [Lyapunov exponents, relatedTo, Lyapunov dimension]
-
A.
Lyapunov exponents
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.
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B.
Hausdorff dimension
The Hausdorff dimension is a mathematical concept in fractal geometry and measure theory that generalizes the notion of dimension to capture the scaling complexity of irregular sets.
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C.
Lyapunov vector
A Lyapunov vector is a mathematical construct in dynamical systems theory that characterizes the directions in phase space associated with exponential growth or decay rates quantified by Lyapunov exponents.
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D.
Lyapunov fractal
The Lyapunov fractal is a complex, self-similar pattern arising from iterating logistic maps with periodically varying parameters, used to visualize stability and chaos in dynamical systems.
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E.
Lyapunov time
Lyapunov time is a measure in dynamical systems theory that quantifies how long it takes for small differences in initial conditions to grow exponentially and significantly affect a system’s future behavior, indicating the timescale of predictability.
- 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 dimension Target entity description: The Lyapunov dimension is a fractal dimension used in dynamical systems theory to quantify the effective number of degrees of freedom of a chaotic attractor based on its Lyapunov exponents.
-
A.
Lyapunov exponents
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.
-
B.
Hausdorff dimension
The Hausdorff dimension is a mathematical concept in fractal geometry and measure theory that generalizes the notion of dimension to capture the scaling complexity of irregular sets.
-
C.
Lyapunov vector
A Lyapunov vector is a mathematical construct in dynamical systems theory that characterizes the directions in phase space associated with exponential growth or decay rates quantified by Lyapunov exponents.
-
D.
Lyapunov fractal
The Lyapunov fractal is a complex, self-similar pattern arising from iterating logistic maps with periodically varying parameters, used to visualize stability and chaos in dynamical systems.
-
E.
Lyapunov time
Lyapunov time is a measure in dynamical systems theory that quantifies how long it takes for small differences in initial conditions to grow exponentially and significantly affect a system’s future behavior, indicating the timescale of predictability.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
chaos quantifier
ⓘ
fractal dimension ⓘ invariant of dynamical systems ⓘ |
| appliesTo |
chaotic attractors
ⓘ
dissipative dynamical systems ⓘ |
| basedOn | Lyapunov exponents ⓘ |
| codomain | non-negative real numbers ⓘ |
| computedFrom | time series via Lyapunov exponent estimation ⓘ |
| dependsOn |
partial sums of ordered Lyapunov exponents
ⓘ
sum of positive Lyapunov exponents ⓘ |
| domain | invariant sets of dynamical systems ⓘ |
| hasAlternativeName |
Kaplan–Yorke Lyapunov dimension
ⓘ
linked to:
Lyapunov dimension
Kaplan–Yorke dimension ⓘ
linked to:
Lyapunov dimension
|
| hasMathematicalNature | real-valued function on invariant sets ⓘ |
| hasProperty |
coordinate invariant
ⓘ
invariant under smooth change of variables ⓘ non-integer in typical chaotic systems ⓘ upper bound for Hausdorff dimension in many systems ⓘ |
| introducedBy |
James A. Yorke
ⓘ
James L. Kaplan ⓘ |
| isDefinedFor |
finite-dimensional dynamical systems
ⓘ
flows ⓘ maps ⓘ |
| isFunctionOf | ordered Lyapunov exponents ⓘ |
| namedAfter | Aleksandr Lyapunov ⓘ |
| quantifies | effective number of degrees of freedom of a chaotic attractor ⓘ |
| relatedTo |
Hausdorff dimension
ⓘ
Kaplan–Yorke conjecture ⓘ
linked to:
Lyapunov dimension
Lyapunov spectrum ⓘ
linked to:
Lyapunov exponents
Pesin theory ⓘ SRB measures ⓘ correlation dimension ⓘ information dimension ⓘ metric entropy ⓘ |
| usedIn |
biological dynamical systems analysis
ⓘ
chaos theory ⓘ climate dynamics ⓘ dynamical systems theory ⓘ engineering control of chaotic systems ⓘ neural dynamics ⓘ nonlinear dynamics ⓘ secure communications based on chaos ⓘ turbulence modeling ⓘ |
| usedToCharacterize |
complexity of chaotic dynamics
ⓘ
strange attractors ⓘ |
| usedToEstimate |
attractor embedding dimension requirements
ⓘ
fractal dimension of attractors ⓘ |
| yearOfIntroduction | 1979 ⓘ |
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 dimension Description of subject: The Lyapunov dimension is a fractal dimension used in dynamical systems theory to quantify the effective number of degrees of freedom of a chaotic attractor based on its Lyapunov exponents.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Lyapunov dimension
linked to: Lyapunov dimension
linked to: Lyapunov dimension