Related papers: Computer Assisted Proofs of Attracting Invariant T…
In a previous work [Asymptotically quasiperiodic solutions for time-dependent Hamiltonians, arXiv preprint arXiv:2211.06623 (2022)], we consider time-dependent perturbations of a Hamiltonian vector field having an invariant torus supporting…
A method to establish regions of phase space through which pass no invariant tori transverse to a given direction field is applied to the planar circular restricted three-body problem. Implications for the location of stable orbits for…
In this work, we introduce a method based on piecewise polynomial interpolation to enclose rigorously solutions of nonlinear ODEs. Using a technique which we call a priori bootstrap, we transform the problem of solving the ODE into one of…
Stationary flows of an inviscid and incompressible fluid of constant density in the region $D=(0, L)\times \mathbb R^2$, periodic in the second and third variables, are considered. The flux and the Bernoulli function are prescribed at each…
We prove reducibility of a transport equation on the $d$-dimensional torus $T^d$ with a time quasi-periodic unbounded perturbation. As far as we know this is the first example of a reducibility result for an equation in more than one…
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…
We consider analytic maps and vector fields defined in $\mathbb{R}^2 \times \mathbb{T}^d$, having a $d$-dimensional invariant torus $\mathcal{T}$. The map (resp. vector field) restricted to $\mathcal{T}$ defines a rotation of frequency…
In the limit of low viscosity, we show that the amplitude of the modes of oscillation of a rotating fluid, namely inertial modes, concentrate along an attractor formed by a periodic orbit of characteristics of the underlying hyperbolic…
In this paper we prove a KAM theorem for small-amplitude solutions of the non linear beam equation on the d-dimensional torus $$u_{tt}+\Delta^2 u+m u + \partial_u G(x,u)=0\ ,\quad t\in { \mathbb{R}} , \; x\in \ { \mathbb{T}}^d, \qquad…
In this article, we discuss the existence and asymptotically autonomous robustness (AAR) (almost surely) of random attractors for 3D stochastic globally modified Navier-Stokes equations (SGMNSE) on Poincar\'e domains (which may be bounded…
We investigate the motion in space of an infinitesimal particle in the gravitational field generated by three primary bodies positioned at the vertices of a fixed equilateral triangle. We assume that the distances between the primaries are…
The existence of global attractors is investigated for the Signorini problem with pointwise dissipation. It is shown that both the semilinear Signorini problem and the elastic obstacle problem with normal compliance exhibit exponential…
We give a comprehensive study of strong uniform attractors of non-autonomous dissipative systems for the case where the external forces are not translation compact. We introduce several new classes of external forces which are not…
We present a deformation for constant mean curvature tori in the 3-sphere. We show that the moduli space of equivariant constant mean curvature tori in the 3-sphere is connected, and we classify the minimal, the embedded, and the Alexandrov…
In this article, we consider the dynamics in a neighborhood of a quasi-periodic torus which is invariant by a Hamiltonian flow, we discuss several notions of stability and we prove several results of instability when the frequency of the…
In this paper we study families of Lagrangian tori that appear in a neighborhood of a resonance of a near-integrable Hamiltonian system. Such families disappear in the "integrable" limit $\varepsilon\to 0$. Dynamics on these tori is…
We prove that a system of coupled nonlinear Schr{\"o}dinger equations on the torus exhibits both stable and unstable small KAM tori. In particular the unstable tori are related to a beating phenomena which has been proved recently in [6].…
Perturbations due to round-off errors in computer modeling are discontinuous and therefore one cannot use results like KAM theory about smooth perturbations of twist maps. We elaborate a special approximation scheme to construct two smooth…
In [3] (Rend. Lincei Mat. Appl. 26 (2015), 1-10; see also arXiv:1503.08145 [math.DS]) the following result has been announced: Theorem. Consider a real-analytic nearly-integrable mechanical system with potential $f$, namely, a Hamiltonian…
At variance from the cases of relative equilibria and relative periodic orbits of dynamical systems with symmetry, the dynamics in relative quasi-periodic tori (namely, subsets of the phase space that project to an invariant torus of the…