Related papers: Computer Assisted Proof of Drift Orbits Along Norm…

We present a topological proof of the existence of a normally hyperbolic invariant manifold for maps. In our approach we do not require that the map is a perturbation of some other map for which we already have an invariant manifold. But a…

We present a computer assisted proof or diffusion in the Planar Elliptic Restricted Three Body Problem. We treat the elliptic problem as a perturbation of the circular problem, where the perturbation parameter is the eccentricity of the…

In this paper we consider a representative a priori unstable Hamiltonian system with 2+1/2 degrees of freedom, to which we apply the geometric mechanism for diffusion introduced in the paper Delshams et al., Mem. Amer. Math. Soc. 2006, and…

We present a general mechanism to establish the existence of diffusing orbits in a large class of nearly integrable Hamiltonian systems. Our approach relies on successive applications of the `outer dynamics' along homoclinic orbits to a…

We present a topological proof of the existence of invariant manifolds for maps with normally hyperbolic-like properties. The proof is conducted in the phase space of the system. In our approach we do not require that the map is a…

We prove the existence of diffusing solutions in the motion of a charged particle in the presence of an ABC magnetic field. The equations of motion are modeled by a 3DOF Hamiltonian system depending on two parameters. For small values of…

We derive quantitative sufficient conditions for rotational chaos and diffusion in annular homeomorphisms, building on the topological criteria established in [31]. These conditions depend only on basic properties of the maps, making their…

Chaotic behavior in dynamical systems poses a significant challenge in trajectory control, traditionally relying on computationally intensive physical models. We present a machine learning-based algorithm to compute the minimum control…

We study the steady states of a system of cross-diffusion equations arising from the modeling of chemotaxis with local sensing, where the motility is a decreasing function of the concentration of the chemical. In order to capture the many…

We show that, in strongly chaotic dynamical systems, the average particle velocity can be calculated analytically by consideration of Brownian dynamics in phase space, the method of images and use of the classical diffusion equation. The…

In the last three decades, powerful computer-assisted techniques have been developed in order to validate a posteriori numerical solutions of semilinear elliptic problems of the form $\Delta u +f(u,\nabla u) = 0$. By studying a well chosen…

We study the problem of Arnold's diffusion in an example of isochronous system by using a geometrical method known as Windows Method. Despite the simple features of this example, we show that the absence of an anisochrony term leads to…

For a given a normally hyperbolic invariant manifold, whose stable and unstable manifolds intersect transversally, we consider several tools and techniques to detect trajectories with prescribed itineraries: the scattering map, the…

We present a new proof of the existence of normally hyperbolic manifolds and their whiskers for maps. Our result is not perturbative. Based on the bounds on the map and its derivative, we establish the existence of the manifold within a…

We introduce a novel technique to find the asymptotic time behaviour of deterministic systems exhibiting anomalous diffusion. The procedure is tested for various classes of simple but physically relevant 1-D maps and possible relevance of…

A new technique for obtaining rigorous results concerning the global dynamics of nonlinear systems is described. The technique combines abstract existence results based on the Conley index theory with computer- assisted computations. As an…

The chaotic diffusion for particles moving in a time dependent potential well is described by using two different procedures: (i) via direct evolution of the mapping describing the dynamics and ; (ii) by the solution of the diffusion…

We present a methodology for computer assisted proofs of Shil'nikov homoclinic intersections. It is based on geometric bounds on the invariant manifolds using rate conditions, and on propagating the bounds by an interval arithmetic…

The rigorous analytical calculation of the diffusion coefficient is performed for the chaotic motion of a particle in a set of longitudinal waves with random phases and large amplitudes (~ A). A first step proves the existence of a…

The scaling invariance for chaotic orbits near a transition from unlimited to limited diffusion in a dissipative standard mapping is explained via the analytical solution of the diffusion equation. It gives the probability of observing a…