An example of Arnold diffusion for near-integrable Hamiltonians

Abstract

Abstract. In this paper, using the ideas of Bessi and Mather, we present a simple mechanical system exhibiting Arnold diffusion. This system of a particle in a small periodic potential can be also interpreted as ray propagation in a periodic optical medium with a near-constant index of refraction. Arnold diffusion in this context manifests itself as an arbitrary finite change of direction for nearly constant index of refraction.

Arnol′d Diffusion in a Pendulum Lattice

“…The same motion, viewed in the configuration space R 4 , is shown in Figure 5. The motion just described is similar to that in a slightly simpler example described in [KL1,KL2]. Stage 2: advance.…”

Arnold Diffusion in A Priori Chaotic Symplectic Maps

“…Indeed, take any z out with z out ≤ δ and consider the mapū σ (u, v, z out )), (62) where z ≤ δ, u ≤ δ and v ∈ A ρ (for some ρ small enough).…”

“…Arnold's paper inspired a large number of studies in the longtime stability of actions, the problem which is known as "Arnold diffusion". It has been attracting significant attention recently and we refer the reader to papers [11,31,33,[40][41][42][43]54,57,[60][61][62][68][69][70]76,82,84] for a more detailed discussion.…”

Viscous stability of quasi-periodic tori

“…Such regularity allows one to introduce a perturbation to make the Mañé sets totally disconnected outside of some neighborhoods of the Mather sets for all P simultaneously, which makes the construction of orbits connecting different Mather sets possible. However, the P−regularity of the Peierls barrier for any P is not generally expected in higher dimensions even in a weak sense (see [19,29]). We assume the following conditions for the Tonelli Lagrangian L: (A1) L(x, v) is of class C 4 (T n × R n ); (A2) its Euler-Lagrange flow admits a quasi-periodic, invariant n-torus…”