Arnold diffusion

In applied mathematics, Arnold diffusion is the phenomenon of instability of integrable Hamiltonian systems. The phenomenon is named after Vladimir Arnold who was the first to publish a result in the field in 1964.[1][2] More precisely, Arnold diffusion refers to results asserting the existence of solutions to nearly integrable Hamiltonian systems that exhibit a significant change in the action variables.

Arnold diffusion describes the diffusion of trajectories due to the ergodic theorem in a portion of phase space unbound by any constraints (i.e. unbounded by Lagrangian tori arising from constants of motion) in Hamiltonian systems. It occurs in systems with more than N=2 degrees of freedom, since the N-dimensional invariant tori do not separate the 2N-1 dimensional phase space any more. Thus, an arbitrarily small perturbation may cause a number of trajectories to wander pseudo-randomly through the whole portion of phase space left by the destroyed tori.

Background and statement

For integrable systems, one has the conservation of the action variables. According to the KAM theorem if we perturb an integrable system slightly, then many, though certainly not all, of the solutions of the perturbed system stay close, for all time, to the unperturbed system. In particular, since the action variables were originally conserved, the theorem tells us that there is only a small change in action for many solutions of the perturbed system.

However, as first noted in Arnold's paper,[1] there are nearly integrable systems for which there exist solutions that exhibit arbitrarily large growth in the action variables. More precisely, Arnold considered the example of nearly integrable Hamiltonian system with Hamiltonian

${\displaystyle H(I,p,q,\phi ,t)={1 \over 2}I^{2}+p^{2}+\epsilon \cos {(q-1)}+\mu \cos {(q-1)}(\sin {\phi +\cos t)}}$

He showed that for this system, with any choice of ${\displaystyle \epsilon ,\delta ,K>0,}$ where ${\displaystyle K\gg \delta }$, there exists a ${\displaystyle \mu _{0}>0}$ such that for all ${\displaystyle \mu \in (0,\mu _{0})}$ there is a solution to the system for which

${\displaystyle I(0)<\delta {\text{ and }}I(T)>K}$

for some time ${\displaystyle T\gg 0.}$

A background on the KAM theorem can be found in [3] and a compendium of rigorous mathmetical results, with insight from physics, can be found in.[4]

See also

• KAM theorem
• Nekhoroshev estimates

References

1. Arnold, Vladimir I. (1964). "Instability of dynamical systems with several degrees of freedom". Soviet Mathematics. 5: 581–585.
2. Florin Diacu; Philip Holmes (1996). Celestial Encounters: The Origins of Chaos and Stability. Princeton University Press. p. 193. ISBN 0-691-00545-1.
3. Henk W. Broer, Mikhail B. Sevryuk (2007) KAM Theory: quasi-periodicity in dynamical systems In: H.W. Broer, B. Hasselblatt and F. Takens (eds.), Handbook of Dynamical Systems Vol. 3, North-Holland, 2010
4. Pierre Lochak, (1999) Arnold diffusion; a compendium of remarks and questions In "Hamiltonian Systems with Three or More Degrees of Freedom" (S’Agar´o, 1995), C. Sim´o, ed, NATO ASI Series C: Math. Phys. Sci., Vol. 533, Kluwer Academic, Dordrecht (1999), 168–183.