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Arnold diffusion for a complete family of perturbations
(Springer, 2017-01-01)
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In this work we illustrate the Arnold diffusion in a concrete example — the a priori unstable Hamiltonian system of 2 + 1/2 degrees of freedom H(p, q, I, f, s) = p2/2+ cos q - 1 + I2/2 + h(q, f, s; e) — proving that for ...
Exponentially small splitting of separatrices associated to 3D whiskered tori with cubic frequencies
(2020-01-01)
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We study the splitting of invariant manifolds of whiskered (hyperbolic) tori with three frequencies in a nearly-integrable Hamiltonian system, whose hyperbolic part is given by a pendulum. We consider a 3-dimensional torus ...
Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation
(2015)
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We consider the cubic defocusing nonlinear Schrödinger equation in the two dimensional torus. Fix s>1. Recently Colliander, Keel, Staffilani, Tao and Takaoka proved the existence of solutions with s -Sobolev norm growing ...
Secular instability in the three-body problem
(2016-02-03)
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© 2016 European Union Consider the three-body problem, in the regime where one body revolves far away around the other two, in space, the masses of the bodies being arbitrary but fixed; in this regime, there are no resonances ...
Splitting of separatrices in the resonances of nearly integrable Hamiltonian Systems of one and a half degrees of freedom
(2013)
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In this paper we consider general nearly integrable analytic Hamiltonian systems of one and a half degrees of freedom which are a trigonometric polynomial in the angular state variable. In the resonances of these systems ...
Instability of high dimensional Hamiltonian systems: Multiple resonances do not impede diffusion
(2016-05-14)
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We consider models given by Hamiltonians of the form View the MathML sourceH(I,f,p,q,t;e)=h(I)+¿j=1n±(12pj2+Vj(qj))+eQ(I,f,p,q,t;e) Turn MathJax on where I¿I¿Rd,f¿TdI¿I¿Rd,f¿Td, p,q¿Rnp,q¿Rn, t¿T1t¿T1. These are higher ...
A general mechanism of diffusion in hamiltonian systems: qualitative results
(2019-01-01)
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We present a general mechanism to establish the existence of diffusing orbits in a large class of nearly integrable Hamiltonian systems. Our approach is based on following the “outer dynamics” along homoclinic orbits to a ...
Exponentially small asymptotic formulas for the length spectrum in some billiard tables
(2016-04-05)
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Let q = 3 be a period. There are at least two (1, q)-periodic trajectories inside any smooth strictly convex billiard table. We quantify the chaotic dynamics of axisymmetric billiard tables close to their boundaries by ...
A parameterization method for Lagrangian tori of exact symplectic maps of R2r
(2018-01-01)
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We are concerned with analytic exact symplectic maps of ${\mathbb R}^{2r}$ endowed with the standard symplectic form. We study the existence of a real analytic torus of dimension $r$, invariant by the map and carrying ...
Critical velocity in kink-defect interaction models: rigorous results
(2020-01-01)
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In this work we study a model of interaction of kinks of the sine-Gordon equation with a weak defect. We obtain rigorous results concerning the so-called critical velocity derived in [7] by a geometric approach. More ...