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Instability of high dimensional hamiltonian systems: Multiple resonances do not impede diffusion
, 2013
"... We consider models given by Hamiltonians of the form n∑ 1 H(I, ϕ, p, q, t; ε) = h(I)+ ± ..."
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We consider models given by Hamiltonians of the form n∑ 1 H(I, ϕ, p, q, t; ε) = h(I)+ ±
Normally Hyperbolic Invariant Laminations and diffusive behaviour for the generalized Arnold . . .
, 2015
"... 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 ..."
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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 ZaslavskiiFilonenko [61] and studied in a multidimensional case by Treschev and Piftankin [51, 52, 55, 56], for an open class of trigonometric perturbations we prove that NHIL do exist. Moreover, using a second order expansion for the separatrix map from [27], we prove that the system restricted to this NHIL is a skew product of nearly integrable cylinder maps. Application of the results from [11] about random iteration of such skew products show that in the proper εdependent time scale the push forward of a Bernoulli measure supported on this NHIL weakly converges to an Ito diffusion process on the line as ε tends to zero.
A second order expansion of the separatrix map for trigonometric perturbations of a priori unstable systems
, 2015
"... ..."
unknown title
, 2015
"... In this paper we propose a model of random compositions of maps of a cylinder, which in the simplified form is as follows: (θ, r) ∈ T×R = A and f±1: ..."
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In this paper we propose a model of random compositions of maps of a cylinder, which in the simplified form is as follows: (θ, r) ∈ T×R = A and f±1:
applications to Arnold diffusion
, 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 introd ..."
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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 two well separated groups and call them dominant. We prove that the associated slow system can be well approximated by a subsystem given by one of the groups, both in the sense of the vector field and weak KAM theory. One of crucial ingredients of proving Arnold diffusion is understanding the structure of invariant (Aubry) sets of nearly integrable systems. As an important application we construct a diffusion path for a generic nearly integrable system such that invariant (Aubry) sets along this path have a "simple " structure similar to the structure of AubryMather sets of twist maps. This is a crucial ingredient in proving Arnold diffusion for convex Hamiltonians in any number of degrees of freedom.