Arnold diffusion
E1041764
Arnold diffusion is a phenomenon in Hamiltonian dynamical systems where very small perturbations can cause slow, long-term drift of trajectories across invariant tori in phase space, leading to global instability.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Arnold diffusion canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T13443969 — 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: Arnold diffusion Context triple: [Kolmogorov–Arnold–Moser theory, relatedTo, Arnold diffusion]
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A.
Kolmogorov–Arnold–Moser theory
Kolmogorov–Arnold–Moser theory is a fundamental result in dynamical systems that explains the persistence of quasi-periodic motions in nearly integrable Hamiltonian systems under small perturbations.
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B.
Liouville–Arnold theorem
The Liouville–Arnold theorem is a fundamental result in Hamiltonian mechanics that guarantees the integrability of a system with sufficiently many conserved quantities and describes its motion as quasi-periodic on invariant tori in phase space.
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C.
Young tower construction in nonuniformly hyperbolic dynamics
"Young tower construction in nonuniformly hyperbolic dynamics" is a foundational work in dynamical systems that introduced a powerful tower-based method for analyzing statistical properties such as decay of correlations and limit theorems in nonuniformly hyperbolic systems.
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D.
Poincaré–Birkhoff fixed-point theorem
The Poincaré–Birkhoff fixed-point theorem is a fundamental result in dynamical systems and topology that guarantees the existence of at least two fixed points for certain area-preserving twist maps of an annulus.
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E.
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.
- 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: Arnold diffusion Target entity description: Arnold diffusion is a phenomenon in Hamiltonian dynamical systems where very small perturbations can cause slow, long-term drift of trajectories across invariant tori in phase space, leading to global instability.
-
A.
Kolmogorov–Arnold–Moser theory
Kolmogorov–Arnold–Moser theory is a fundamental result in dynamical systems that explains the persistence of quasi-periodic motions in nearly integrable Hamiltonian systems under small perturbations.
-
B.
Liouville–Arnold theorem
The Liouville–Arnold theorem is a fundamental result in Hamiltonian mechanics that guarantees the integrability of a system with sufficiently many conserved quantities and describes its motion as quasi-periodic on invariant tori in phase space.
-
C.
Young tower construction in nonuniformly hyperbolic dynamics
"Young tower construction in nonuniformly hyperbolic dynamics" is a foundational work in dynamical systems that introduced a powerful tower-based method for analyzing statistical properties such as decay of correlations and limit theorems in nonuniformly hyperbolic systems.
-
D.
Poincaré–Birkhoff fixed-point theorem
The Poincaré–Birkhoff fixed-point theorem is a fundamental result in dynamical systems and topology that guarantees the existence of at least two fixed points for certain area-preserving twist maps of an annulus.
-
E.
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.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
dynamical systems phenomenon
ⓘ
phenomenon in Hamiltonian dynamical systems ⓘ |
| causes |
migration of orbits across KAM tori gaps
ⓘ
slow variation of action variables ⓘ |
| characterizedBy |
exponentially small effects in perturbation size
ⓘ
extremely slow instability ⓘ non-local transport in phase space ⓘ |
| contrastsWith | quasi-periodic motion on invariant tori ⓘ |
| describes |
global instability in nearly integrable Hamiltonian systems
ⓘ
slow drift of trajectories in phase space ⓘ |
| field |
Hamiltonian dynamics
ⓘ
dynamical systems theory ⓘ mathematical physics ⓘ |
| firstProposedBy | Vladimir Igorevich Arnold NERFINISHED ⓘ |
| firstProposedIn | 1964 ⓘ |
| formalizedAs | existence of orbits with unbounded drift in action variables ⓘ |
| hasOpenProblemsIn |
genericity and prevalence in concrete physical models
ⓘ
numerical detection and verification ⓘ rigorous estimates of diffusion speed ⓘ |
| hasProperty |
difficult to observe numerically due to long timescales
ⓘ
generic in sufficiently high-dimensional Hamiltonian systems ⓘ |
| implies | lack of global stability in nearly integrable Hamiltonian systems ⓘ |
| involves |
motion along resonance channels in phase space
ⓘ
slow, long-term drift across invariant tori ⓘ transition between different regions of action space ⓘ very small perturbations of integrable Hamiltonian systems ⓘ |
| mechanism |
chains of connected resonances
ⓘ
overlap of resonances in high-dimensional phase space ⓘ |
| modeledBy | perturbed integrable Hamiltonians ⓘ |
| namedAfter | Vladimir Igorevich Arnold NERFINISHED ⓘ |
| occursIn |
Hamiltonian systems with three or more degrees of freedom
ⓘ
nearly integrable Hamiltonian systems ⓘ |
| relatedTo |
Hamiltonian chaos
ⓘ
KAM tori ⓘ Kolmogorov–Arnold–Moser theory NERFINISHED ⓘ Nekhoroshev theory NERFINISHED ⓘ instability of nearly integrable systems ⓘ resonant tori ⓘ |
| requires |
at least three degrees of freedom
ⓘ
breakdown or partial destruction of invariant tori ⓘ presence of a web of resonances ⓘ |
| studiedIn |
accelerator physics
ⓘ
astrodynamics ⓘ celestial mechanics ⓘ plasma physics ⓘ |
| timeScale | can be exponentially long in inverse perturbation size ⓘ |
| typicalSetting | analytic or smooth Hamiltonian systems ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Arnold diffusion Description of subject: Arnold diffusion is a phenomenon in Hamiltonian dynamical systems where very small perturbations can cause slow, long-term drift of trajectories across invariant tori in phase space, leading to global instability.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.