Related papers: Kolmogorov algorithm for isochronous Hamiltonian s…
We give a simple proof of Kolmogorov's theorem on the persistence of a quasiperiodic invariant torus in Hamiltonian systems. The theorem is first reduced to a well-posed inversion problem (Herman's normal form) by switching the frequency…
We give a proof of the convergence of an algorithm for the construction of lower dimensional elliptic tori in nearly integrable Hamiltonian systems. The existence of such invariant tori is proved by leading the Hamiltonian to a suitable…
We investigate the long-term dynamics of HD60532, an extrasolar system hosting two giant planets orbiting in a 3:1 mean motion resonance. We consider an average approximation at order one in the masses which results (after the reduction of…
We give a constructive proof of the existence of lower dimensional elliptic tori in nearly integrable Hamiltonian systems. In particular we adapt the classical Kolmogorov's normalization algorithm to the case of planetary systems, for which…
In this paper, it is proved that the infinite KAM torus with prescribed frequency exists in a sufficiently small neighborhood of a given $ I^{0}$ for nearly integrable and analytic Hamiltonian system $ H(I,\theta) = H_{0}(I)+ \epsilon…
A universal algorithm to construct N-particle (classical and quantum) completely integrable Hamiltonian systems from representations of coalgebras with Casimir element is presented. In particular, this construction shows that quantum…
In this paper, we investigate Kolmogorov type theorems for small perturbations of degenerate Hamiltonian systems. These systems are index by a parameter $\xi$ as \( H(y,x,\xi) = \langle\omega(\xi),y\rangle + \varepsilon…
Consider an analytic Hamiltonian system near its analytic invariant torus $\mathcal T_0$ carrying zero frequency. We assume that the Birkhoff normal form of the Hamiltonian at $\mathcal T_0$ is convergent and has a particular form: it is an…
In this paper we reconsider the original Kolmogorov normal form algorithm with a variation on the handling of the frequencies. At difference with respect to the Kolmogorov approach, we do not keep the frequencies fixed along the…
In this paper we present an a-posteriori KAM theorem for the existence of an $(n-d)$-parameters family of $d$-dimensional isotropic invariant tori with Diophantine frequency vector $\omega\in \mathbb R^d$, of type $(\gamma,\tau)$, for $n$…
We consider the classical problem of the construction of invariant tori exploiting suitable Hamiltonian normal forms. This kind of approach can be translated by means of the Lie series method into explicit computational algorithms, which…
We prove an abstract Birkhoff normal form theorem for Hamiltonian partial differential equations on torus. The normal form is complete up to arbitrary finite order. The proof is based on a valid non-resonant condition and a suitable norm of…
The Hamiltonian approach to isomonodromic deformation systems for generic rational covariant derivative operators on the Riemann sphere, having any matrix dimension $r$ and any number of isolated singularities of arbitrary Poincar\'e rank,…
In this paper we develop a straightforward procedure to construct higher dimensional isochronous Hamiltonian systems. We first show that a class of singular Hamiltonian systems obtained through the $\Omega$-modified procedure is equivalent…
In this paper we present efficient algorithms for the computation of several invariant objects for Hamiltonian dynamics. More precisely, we consider KAM tori (i.e diffeomorphic copies of the torus such that the motion on them is conjugated…
We adapt the Kolmogorov's normalization algorithm (which is the key element of the original proof scheme of the KAM theorem) to the construction of a suitable normal form related to an invariant elliptic torus. As a byproduct, our procedure…
The monodromy of torus bundles associated to completely integrable systems can be computed using geometric techniques (constructing homology cycles) or analytic arguments (computing discontinuities of abelian integrals). In this article we…
We consider a class of perturbations of the 2D harmonic oscillator, and of some other dynamical systems, which we show are isomorphic to a function of a toric system (a Birkhoff canonical form). We show that for such systems there exists a…
This paper is devoted to studying Hamiltonian oscillators in 1:1:1:1 resonance with symmetries, which include several models of perturbed Keplerian systems. Normal forms are computed in Poisson and symplectic formalisms, by mean of…
In this article we formulate and prove sufficient conditions for the existence of trajectories of nonstationary periodic solutions of autonomous Hamiltonian systems in a neighbourhood of equilibria. It is worth pointing out that assumptions…