Related papers: Closing lemma and KAM normal form

Our understanding of the mechanisms governing the structure and secular evolution galaxies assume nearly integrable Hamiltonians with regular orbits; our perturbation theories are founded on the averaging theorem for isolated resonances. On…

Integrable Hamiltonian systems on almost-symplectic manifolds have recently drawn some attention. Under suitable properties, they have a structure analogous to those of standard symplectic-Hamiltonian completely integrable systems. Here we…

We study the dynamics of solutions for a family of nonlinear Schroedinger equations on the circle, with a smooth convolution potential and Gevrey regular initial data. Our main result is the construction of an asymptotically full measure…

Consider a sufficiently smooth nearly integrable Hamiltonian system of two and a half degrees of freedom in action-angle coordinates \[ H_\epsilon (\varphi,I,t)=H_0(I)+\epsilon H_1(\varphi,I,t), \varphi\in T^2,\ I\in U\subset R^2,\ t\in…

We consider Gevrey perturbations $H$ of a completely integrable Gevrey Hamiltonian $H_0$. Given a Cantor set $\Omega_\kappa$ defined by a Diophantine condition, we find a family of KAM invariant tori of $H$ with frequencies $\omega\in…

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 Cantor families of small amplitude, linearly stable, quasi-periodic solutions of quasi-linear autonomous Hamiltonian generalized KdV equations. We consider the most general quasi-linear quadratic nonlinearity. The…

We consider a $C^{1,\alpha}$ smooth flow in $\mathbb{R}^n$ which is "strongly monotone" with respect to a cone $C$ of rank $k$, a closed set that contains a linear subspace of dimension $k$ and no linear subspaces of higher dimension. We…

The main result in this paper is the $C^{\infty}$ closing lemma for a large family of Hamiltonian flows on $4$-dimensional symplectic manifolds, which includes classical Hamiltonian systems. First we prove the $C^{\infty}$ closing lemma and…

Written with respect to an appropriate Poisson structure, a partially integrable Hamiltonian system is viewed as a completely integrable system with parameters. Then, the theorem on quasi-periodic stability in Ref. [1] (the KAM theorem) can…

Poincar\'e recurrence theorem implies the density of recurrent points for volume-preserving dynamical systems on compact domains. The density of closed orbits in the non-wandering set is one of the essential properties of Axiom A and chaos.…

We compute the normal forms for the Hamiltonian leading to the epicyclic approximations of the (perturbed) Kepler problem in the plane. The Hamiltonian setting corresponds to the dynamics in the Hill synodic system where, by means of the…

Beyond H\"{o}lder's type, this paper mainly concerns the persistence and remaining regularity of an individual frequency-preserving KAM torus in a finitely differentiable Hamiltonian system, even allows the non-integrable part being…

In \cite{GL21a} we have developed a new approach to $L^{\infty}$-a priori estimates for degenerate complex Monge-Amp\`ere equations, when the reference form is closed. This simplifying assumption was used to ensure the constancy of the…

We consider the $ C^1 $ linearization of a perturbed vector field $ \omega+P $ over the infinite-dimensional torus $ \mathbb{T}^\infty $, and determine the sharp regularity requirement of the perturbation $ P $ conjugating the unperturbed…

We give new estimates for a critical elliptic system introduced by Rivi\`ere-Struwe in \cite{riviere_struwe} (see also the work of Rupflin \cite{rupflin} and Schikorra \cite{schikorra_frames}), which generalises PDE solved by harmonic (and…

We review a recent generalization of Normal Form Theory to systems (Hamiltonian ones or general ODEs) where the perturbing term is not periodic in one coordinate variable. The main difference with the standard case relies on the non…

This work proposes a new way for handling obstacles to asymptotic integrability in perturbed nonlinear PDEs within the method of Normal Forms - NF - for the case of multi-wave solutions. Instead of including the whole obstacle in the NF,…

We use the homological perturbation lemma to produce explicit formulas computing the class in the twisted de Rham complex represented by an arbitrary polynomial. This is a non-asymptotic version of the method of Feynman diagrams. In…

We consider a geodesic flow on a compact manifold endowed with a Riemannian (or Finsler, or Lorentz) metric satisfying some generic, explicit conditions. We couple the geodesic flow with a time-dependent potential, driven by an external…