Related papers: Biasymptotically quasiperiodic solutions for time-…
We prove the existence and stability of Cantor families of quasi-periodic, small amplitude solutions of quasi-linear (i.e. strongly nonlinear) autonomous Hamiltonian perturbations of KdV.
In this paper, we prove the existence of quasi-periodic small-amplitude solutions for quasi-linear Hamiltonian perturbation of the fifth-order KdV equation on the torus in presence of a quasi-periodic forcing.
We consider Hamiltonians associated to optimal control problems for affine systems on the torus. They are not coercive and are possibly unbounded from below in the direction of the drift of the system. The main assumption is the strong…
We prove that there is an invariant torus with given Diophantine frequency vector for a class of Hamiltonian systems defined by an integrable large Hamiltonian function with a large non-autonomous Hamiltonian perturbation. As for…
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 existence of whiskered tori in some dissipative systems, called \sl conformally symplectic \rm systems, having the property that they transform the symplectic form into a multiple of itself. We consider a family $f_\mu$…
Rotators interacting with a pendulum via small, velocity independent, potentials are considered. If the interaction potential does not depend on the pendulum position then the pendulum and the rotators are decoupled and we study the…
We analyze the breakup of invariant tori in Hamiltonian systems with two degrees of freedom using a combination of KAM theory and renormalization-group techniques. We consider a class of Hamiltonians quadratic in the action variables that…
Quasi-periodic trajectories with two or more incommensurate frequencies are ubiquitous in nonlinear dynamics, yet the classical Fourier-based time-spectral method is tied to strictly periodic responses. We introduce a torus time-spectral…
We study the existence of whiskered tori in a family $f_\mu$ of conformally symplectic maps depending on parameters $\mu$. Whiskered tori are tori on which the motion is a rotation, but they have as many expanding/contracting directions as…
When a Gevrey smooth perturbation is applied to a quasi-convex integrable Hamiltonian, it is known that the KAM invariant tori that survive are sticky, that is, doubly exponentially stable. We show by examples the optimality of this…
We present a set of time quasi-periodic solutions to a nonlinear Klein-Gordon equation with a decaying nonlinear term on the torus in arbitrary dimensions. This paper generalizes the bifurcation method developed in [W2].
We consider one-dimensional systems in the presence of a quasi-periodic perturbation, in the analytical setting, and study the problem of existence of quasi-periodic solutions which are resonant with the frequency vector of the…
We work with small non-selfadjoint perturbations of a selfadjoint quantum Hamiltonian with two degrees of freedom, assuming that the principal symbol of the selfadjoint part is (classically) a nearly integrable system, together with a…
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…
We consider a singular perturbation for a family of analytic symplectic maps of the annulus possessing a KAM torus. The perturbation introduces dissipation and contains an adjustable parameter. By choosing the adjustable parameter, one can…
We study a Hamiltonian describing a pendulum coupled with several anisochronous oscillators, giving a simple construction of unstable KAM tori and their stable and unstable manifolds for analytic perturbations. We extend analytically 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 this paper, we survey our recent results on the Benjamin-Ono equation on the torus. As an application of the methods developed we construct large families of periodic or quasiperiodic solutions, which are not $C^\infty$-smooth.
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…