1,721,049 research outputs found
On the measure of kam tori in two degrees of freedom
No full textA conjecture of Arnold, Kozlov and Neishtadt on the exponentially small measure of the “non-torus” set in analytic systems with two degrees of freedom is discussed
Global properties of generic real–analytic nearly–integrable Hamiltonian systems
No full textWe introduce a new class Gns of generic real analytic potentials on Tn and study global analytic properties of natural nearly-integrable Hamiltonians 12|y|2 + epsilon f (x), with potential f is an element of Gns, on the phase space M = B x Tn with B a given ball in Rn. The phase space M can be covered by three sets: a 'non-resonant' set, which is filled up to an exponentially small set of measure e-cK (where K is the maximal size of resonances considered) by primary maximal KAM tori; a 'simply resonant set' of measure root epsilon Ka and a third set of measure epsilon Kb which is 'non perturbative', in the sense that the H-dynamics on it can be described by a natural system which is not nearly-integrable. We then focus on the simply resonant set - the dynamics of which is particularly interesting (e.g., for Arnol'd diffusion, or the existence of secondary tori) - and show that on such a set the secular (averaged) 1 degree-of-freedom Hamiltonians (labeled by the resonance index k is an element of Zn) can be put into a universal form (which we call 'Generic Standard Form'), whose main analytic properties are controlled by only one parameter, which is uniform in the resonance label k. (c) 2023 Elsevier Inc. All rights reserved
On the topology of nearly-integrable Hamiltonians at simple resonances
No full textWe show that, in general, averaging at simple resonances a real-analytic, nearly-integrable Hamiltonian, one obtains a one-dimensional system with a cosine-like potential; 'in general' means for a generic class of holomorphic perturbations and apart from a finite number of simple resonances with small Fourier modes; 'cosine-like' means that the potential depends only on the resonant angle, with respect to which it is a Morse function with one maximum and one minimum. Furthermore, the (full) transformed Hamiltonian is the sum of an effective one-dimensional Hamiltonian (which is, in turn, the sum of the unperturbed Hamiltonian plus the cosine-like potential) and a perturbation, which is uniformly exponentially small. As a corollary, under the above hypotheses, if the unperturbed Hamiltonian is also strictly convex, the effective Hamiltonian at any simple resonance (apart a finite number of low-mode resonances) has the phase portrait of a pendulum. The results presented in this paper are an essential step in the proof (in the 'mechanical' case) of a conjecture by Arnold-Kozlov-Neishdadt [Arnold V I, Kozlov V V and Neishtadt A I 2006 Mathematical aspects of classical and celestial mechanics Encyclopaedia of Mathematical Sciences 3rd edn vol 3 (Berlin: Springer), remark 6.8, p 285], claiming that the measure of the 'non-torus set' in general nearly-integrable Hamiltonian systems has the same size of the perturbation; compare [Biasco L and Chierchia L 2015 On the measure of Lagrangian invariant tori in nearly-integrable mechanical systems Rendiconti Lincei. Mat. Appl. 26 1-10 and Biasco L and Chierchia L KAM Theory for Secondary Tori (arXiv:1702.06480v1 [math.DS])]
Periodic solutions of Birkhoff-Lewis type for the nonlinear wave equation
No full textWe prove the existence of infinitely many periodic solutions accumulating to zero for the one-dimensional nonlinear wave equation (vibrating string equation). The periods accumulate to zero and are both rational and irrational multiples of the string length
Time periodic solutions for the nonlinear wave equation with long minimal period
No full textWe prove existence and multiplicity of small amplitude periodic solutions for the wave equation with small "mass" and odd nonlinearity. Such solutions bifurcate from resonant finite dimensional invariant tori of the fourth order Birkhoff normal form of the associated Hamiltonian system. The number of geometrically distinct solutions and their minimal periods go to infinity when the "mass" goes to zero. This is the first result about long minimal period for the autonomous wave equation. © 2006 Society for Industrial and Applied Mathematics
Periodic solutions of Birkhoff-Lewis type for the nonlinear wave equation
No full textWe prove existence and multiplicity of small amplitude periodic solutions with large period for the wave equation with small "mass". Such solutions bifurcate from resonant finite-dimensional invariant tori of the fourth order Birkhoff normal form of the associated hamiltonian system. The number of geometrically distinct solutions and their minimal periods tend to infinity when the "mass" tends to zero
A note on the construction of Sobolev almost periodic invariant tori for the 1d NLS
Full text linkWe announce a method for the construction of almost periodic solutions of the one dimensional analytic NLS with only Sobolev regularity both in time and space. This is the first result of this kind for PDEs
Periodic orbits close to elliptic tori and applications to three body problem
No full textWe prove, under suitable non-resonance and non-degeneracy “twist” conditions, a Birkhoff-Lewis type result showing the existence of infinitely many periodic solutions, with larger and larger minimal period, accumulating onto elliptic invariant tori (of Hamiltonian systems). We prove the applicability of this result to the spatial planetary three-body problem in the small eccentricity-inclination regime. Furthermore, we find other periodic orbits under some restrictions on the period and the masses of the “planets”. The proofs are based on averaging theory, KAM theory and variational methods
KAM theory for the Hamiltonian derivative wave equation
No full textWe prove an infinite dimensional KAM theorem which implies the existence of Cantor families of small-amplitude, reducible, elliptic, analytic, invariant tori of Hamiltonian derivative wave equations. A key concept is the introduction of the class of quasi-Toplitz Hamiltonians, which provides a sharp asympototic decay estimate for the eigenvalues of the linearized operators at each KAM step
An Abstract Birkhoff Normal Form Theorem and Exponential Type Stability of the 1d NLS
Full text linkWe study stability times for a family of parameter dependent nonlinear Schrödinger equations on the circle, close to the origin. Imposing a suitable Diophantine condition (first introduced by Bourgain), we prove a rather flexible Birkhoff Normal Form theorem, which implies, e.g., exponential and sub-exponential time estimates in the Sobolev and Gevrey class respectively
- …
