Arnold diffusion

E1041764

dynamical systems phenomenon phenomenon in Hamiltonian dynamical systems

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.

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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 ⓘ

Referenced by (2)

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

Vladimir Arnold → knownFor → Arnold diffusion ⓘ

Kolmogorov–Arnold–Moser theory → relatedTo → Arnold diffusion ⓘ