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- V Kaloshin, K Zhang
- 2014

In the present paper we prove a strong form of Arnold diffusion. Let T 2 be the two torus and B 2 be the unit ball around the origin in R 2. Fix ρ > 0. Our main result says that for a " generic " time-periodic perturbation of an integrable system of two degrees of freedom H 0 (p) + εH 1 (θ, p, t), θ ∈ T 2 , p ∈ B 2 , t ∈ T = R/Z, with a strictly convex H 0… (More)

In the present paper we prove a form of Arnold diffusion. The main result says that for a " generic " perturbation of a nearly integrable system of arbitrary degrees of freedom n 2 H 0 (p) + εH 1 (θ, p, t), θ ∈ T n , p ∈ B n , t ∈ T = R/T, with strictly convex H 0 there exists an orbit (θ ǫ , p e)(t) exhibiting Arnold diffusion in the sens that sup t>0 p(t)… (More)

- Vadim Kaloshin, Ke Zhang, Yong Zheng
- 2011

We study a C r nearly integrable Hamiltonian system Hε(q, p) = 1 2 p, p + H 1 (q, p) defined on T 3 × R 3. Let Σ = {(q, p) : Hε(q, p) = 1 2 } and µ Σ 1 be the restriction of Lebesgue measure on T 3 × R 3 to Σ. We prove there is a perturbation H 1 (q, p) ∈ C r , H 1 C r ≤ 1 and an orbit (q(t), p(t)) : R → T 3 × R 3 of the Hamiltonian equation { ˙ q = ∂pHε, ˙… (More)

- V Kaloshin, K Zhang
- 2012

In the present paper we consider a generic perturbation of a nearly integrable system of n and a half degrees of freedom with a strictly convex H 0. For n = 2 we show that at a strong double resonance there exist 3-dimensional normally hyperbolic invariant cylinders going across. This is somewhat unexpected, because at a strong double resonance dynamics can… (More)

Let (x, y) be a specified arc in a k-regular bipartite tournament B. We prove that there exists a cycle C of length four through (x, y) in B such that B-C is hamiltonian.

- V Kaloshin, K Zhang
- 2015

It is well know that instabilities of nearly integrable Hamiltonian systems occur around resonances. Dynamics near resonances of these systems is well approximated by the associated averaged system, called slow system. Each resonance is defined by a basis (a collection of integer vectors). We introduce a class of resonances whose basis can be divided into… (More)

- V Kaloshin, K Zhang
- 2014

We present key elements of a proof of a strong form of Arnold diffusion for systems of three and a half degrees of freedom. More exactly, let T 3 be a 3-dimensional torus and B 3 be the unit ball around the origin in R 3. Fix ρ > 0. Our main result says that for a " generic " time-periodic perturbation of an integrable system of three degrees of freedom H 0… (More)

In this paper we study existence of Normally Hyperbolic Invariant Laminations (NHIL) for a nearly integrable system given by the product of the pendulum and the rotator perturbed with a small coupling between the two. This example was introduced by Arnold [1]. Using a separatrix map, introduced in a low dimensional case by Zaslavskii-Filonenko [61] and… (More)