相关论文: Normal forms for perturbations of systems possessi…

We prove a discrete time analogue of 1967 Moser's normal form of real analytic perturbations of vector fields possessing an invariant, reducible, Diophantine torus; in the case of diffeomorphisms too, the persistence of such an invariant…

In this paper, we use geometry of numbers to relate two dual Diophantine problems. This allows us to focus on simultaneous approximations rather than small linear forms. As a consequence, we develop a new approach to the perturbation theory…

We prove the existence of real analytic Hamiltonians with topologically unstable quasi-periodic invariant tori. Using various versions of our examples, we solve the following problems in the stability theory of analytic quasi-periodic…

We consider a class of quasi-integrable Hamiltonian systems obtained by adding to a non-convex Hamiltonian function of an integrable system a perturbation depending only on the angle variables. We focus on a resonant maximal torus of the…

The classical theorem of Moser, on the existence of a normal form in the neighbourhood of a hyperbolic equilibrium, is extended to a class of real-analytic Hamiltonians with aperiodically time-dependent perturbations. A stronger result is…

We study a new type of normal form at a critical point of an analytic Hamiltonian. Under a Bruno condition on the frequency, we prove a convergence statement to the normal form. Using this result, we prove the Herman invariant tori…

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 generalize the Weinstein-Moser theorem on the existence of nonlinear normal modes (i.e., periodic orbits) near an equilibrium in a Hamiltonian system to a theorem on the existence of relative periodic orbits near a relative equilibrium…

Some scales of spaces of ultra-differentiable functions are introduced, having good stability properties with respect to infinitely many derivatives and compositions. They are well-suited for solving non-linear functional equations by means…

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…

A Gevrey symplectic normal form of an analytic and more generally Gevrey smooth Hamiltonian near a Lagrangian invariant torus with a Diophantine vector of rotation is obtained. The normal form implies effective stability of the…

We use Moser's normal forms to study chaotic motion in two-degree hamiltonian systems near a saddle point. Besides being convergent, they provide a suitable description of the cylindrical topology of the chaotic flow in that vicinity. Both…

Moser's invariant tori for a class of nonanalytic quasi integrable even hamiltonian systems are shown to be analytic in the perturbation parameter. We do so by exhibiting a summation rule for the divergent series (``Lindstedt series") that…

We study several aspects of the regular deformations of completely integrable systems. Namely, we prove the existence of a Hamiltonian normal form for these deformations and we show the necessary and sufficient conditions a perturbation has…

In this paper we discuss a class of normal forms of the completely resonant non--linear Schr\"odinger equation on a torus. We stress the geometric and combinatorial constructions arising from this study. Further analytic considerations and…

In this paper, we give a new proof of the classical KAM theorem on the persistence of an invariant quasi-periodic torus, whose frequency vector satisfies the Bruno-R\"ussmann condition, in real-analytic non-degenerate Hamiltonian systems…

Motivated by the Lagrange top coupled to an oscillator, we consider the quasi-periodic Hamiltonian Hopf bifurcation. To this end, we develop the normal linear stability theory of an invariant torus with a generic (i.e., non-semisimple)…

We consider the classical problem of the continuation of periodic orbits surviving to the breaking of invariant lower dimensional resonant tori in nearly integrable Hamiltonian systems. In particular we extend our previous results…

We prove a new invariant torus theorem, for $\alpha$-Gevrey smooth Hamiltonian systems, under an arithmetic assumption which we call the $\alpha$-Bruno-R\"ussmann condition, and which reduces to the classical Bruno-R\"ussmann condition in…

In this paper we adapt the method of [P. H. Baptistelli, M. Manoel and I. O. Zeli. Normal form theory for reversible equivariant vector fields. Bull. Braz. Math. Soc., New Series 47 (2016), no. 3, 935-954] to obtain normal forms of a class…