Arnold diffusion far from strong resonances in multidimensionala prioriunstable Hamiltonian systems

Abstract

We prove the existence of Arnold diffusion in a typical a priori unstable Hamiltonian system outside a small neighbourhood of strong resonances. More precisely, we consider a near-integrable Hamiltonian system with Hamiltonian H = H 0 + εH 1 + O(ε 2 ), where the unperturbed Hamiltonian H 0 is essentially the product of a one-dimensional pendulum and n-dimensional rotator. Coordinates y = (y 1 , . . . , y n ) on the rotator space are first integrals in the unperturbed system and become slow variables after perturbation.The main result is as follows. Suppose that the time-periodic perturbation H 1 is C r -generic, r ∈ N ∪ {∞, ω} is sufficiently large. A resonance k, ν = 0, where ν = ν(y) ∈ R n+1 is a frequency vector and k ∈ Z n+1 , is called strong if |k| < C . The constant C is determined by H 0 and H 1 and does not depend on ε. Let Q be a domain on the rotator space such that its closure Q is free from strong resonances and let γ be a smooth curve on Q. Then for any small ε > 0 there exists a trajectory whose projection to Q moves in a c| log ε| α ε 1/4neighbourhood of γ (α n 2 +2n r−n−1 is any constant and c = c(α) > 0) with average velocity along γ of order ε/| log ε|.

Variational construction of positive entropy invariant measures of Lagrangian systems and Arnold diffusion

“…(9) The "fast diffusion" (with respect to the perturbation µ) for sufficiently small µ has the drift acceleration O(µ/| log µ|), as conjectured by Lochak, and proved for a class of a-priori unstable systems similar to (1.1) (also allowing the dimension of the rotator variable v to be ≥ 1) with non-transveral intersection of whiskers by Berti, Biasco, Bolle, and Treschev [10,45,46]. In [10], it was established that this is under certain assumptions the largest possible drift acceleration.…”

Arnold Diffusion in A Priori Chaotic Symplectic Maps

“…Inequalities (88) and (89) are strict and involve only the first derivatives of the scattering maps. A C 2 -small change of the family Φ (1) μ 1 leads to a C 1 -small change of the strong-stable and strong-unstable foliations and, therefore, a C 1 -small change of the scattering maps.…”

“…A C 2 -small change of the family Φ (1) μ 1 leads to a C 1 -small change of the strong-stable and strong-unstable foliations and, therefore, a C 1 -small change of the scattering maps. Thus, it is enough to build a C 2 -smooth family of maps X (1) μ 1 [generated by a C 3 -smooth Hamiltonian H (1) ] such that the corresponding scattering maps satisfy (88) and (89). Then for any sufficiently C 3 -close approximation of H (1) by an analytic Hamiltonian [the analiticity of H (1) and H (2) is needed for the family Φ μ to be analytic, i.e.…”

“…A system of this type is called a priori unstable. The drift of orbits along the cylinder has been actively studied in the last decade [3][4][5][7][8][9]12,[18][19][20]22,27,28,[31][32][33][34]70,87,88], including the problem of genericity of this phenomenon and instability times. It should be noted that the Arnold diffusion can be much faster in this case.…”

Geometric proof of strong stable/unstable manifolds with application to the restricted three body problem

“…Combined with Melnikov type arguments, these can be used in proofs of Arnold diffusion [2] type dynamics. A broad selection of papers has used this approach, including the work of Delshams, Huguet, de la Llave, Seara or Treschev [13,14,15,16,24,25] amongst many others. Such approach has also been applied in [10], in the setting of the planar elliptic restricted three body problem.…”