Related papers: KAM-tori near an analytic elliptic fixed point
We show that an analytic invariant torus $\cT_0$ with Diophantine frequency $\o_0$ is never isolated due to the following alternative. If the Birkhoff normal form of the Hamiltonian at $\cT_0$ satisfies a R\"ussmann transversality…
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…
For a fixed frequency vector $\omega \in \mathbb{R}^2 \, \setminus \, \lbrace 0 \rbrace$ obeying $\omega_1 \omega_2 < 0$ we show the existence of Gevrey-smooth Hamiltonians, arbitrarily close to an integrable Kolmogorov non-degenerate…
In this article, we consider solutions starting close to some linearly stable invariant tori in an analytic Hamiltonian system and we prove results of stability for a super-exponentially long interval of time, under generic conditions. The…
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 consider linearly stable elliptic fixed points for a symplectic vector field and prove generic results of super-exponential stability for nearby solutions. Morbidelli and Giorgilli have proved a theorem of stability over…
In this short note, we prove that a quasi-periodic torus, with a non-resonant frequency (that can be Diophantine or Liouville) and which is invariant by a sufficiently regular Hamiltonian flow, is KAM stable provided it is Kolmogorov…
In the Nineties, Michel Herman conjectured the existence of a positive measure set of invariant tori at an elliptic diophatine critical point of a hamiltonian function. I construct a formalism for the UV-cutoff and prove a generalised KAM…
In this paper, we present two infinite-dimensional KAM theorems with frequency-preserving for a nonresonant frequency of Diophantine type or even weaker. To be more precise, under a nondegenerate condition for an infinite-dimensional…
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…
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 this paper, we investigate the existence of KAM tori for an infinite dimensional Hamiltonian system with finite number of zero normal frequencies. By constructing a constant quantity we show that, for "most" frequencies in the sense of…
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)…
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…
We construct on $\R^{2d}$, for any $d \geq 3$, smooth Hamiltonians having an elliptic equilibrium with an arbitrary frequency, that is not accumulated by a positive measure set of invariant tori. For $d\geq 4$, the Hamiltonians we construct…
It was conjectured by Herman that an analytic Lagrangian Diophantine quasi-periodic torus $\mathcal{T}_0$, invariant by a real-analytic Hamiltonian system, is always accumulated by a set of positive Lebesgue measure of other Lagrangian…
In this paper, we establish a KAM-theorem for ordinary differential equations with finitely differentiable vector fields and multiple degeneracies. The theorem can be used to deal with the persistence of quasi-periodic invariant tori in…
In the nineties, Michel Herman conjectured the existence of a positive measure set of invariant tori at an elliptic diophantine critical point of a hamiltonian function. I show that KAM versal deformation theory solves positively this…
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 $n$…
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…