Related papers: Efficient manifold tracing for planar maps
We develop a theoretical framework for computer-assisted proofs of the existence of invariant objects in semilinear PDEs. The invariant objects considered in this paper are equilibrium points, traveling waves, periodic orbits and invariant…
The existence of translated curves for quasiperiodically forced maps is established, under very mild regularity hypotheses, for rotation numbers of constant type. Among the translated curves, the invariant curves are characterized as the…
Last two decades, the problem of robotic mapping has made a lot of progress in the research community. However, since the data provided by the sensor still contains noise, how to obtain an accurate map is still an open problem. In this…
We propose a novel framework for approximating the statistical properties of turbulent flows by combining variational methods for the search of unstable periodic orbits with resolvent analysis for dimensionality reduction. Traditional…
Invariant manifolds are fundamental tools for describing and understanding nonlinear dynamics. In this paper, we present a theory of stable and unstable manifolds for infinite dimensional random dynamical systems generated by a class of…
The search of high-order periodic orbits has been typically restricted to problems with symmetries that help to reduce the dimension of the search space. Well-known examples include reversible maps with symmetry lines. The present work…
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…
The direct computation of the third-order normal form for a geometrically nonlinear structure discretised with the finite element (FE) method, is detailed. The procedure allows to define a nonlinear mapping in order to derive accurate…
We present a rigorous numerical scheme for the approximation of the linear response of the invariant density of a map with an indifferent fixed point, with explicit and computed estimates for the error and all the involved constants.
Planar curves with periodically varying curvature arise in the natural sciences as the result of a wide variety of periodic processes. The total curvature of a periodic arc in such curves constrains their symmetry. It is shown how the total…
Numerical N-body simulations are commonly used to explore stability regions around exoplanets, offering insights into the possible existence of satellites and ring systems. This study aims to utilize Machine Learning (ML) techniques to…
It is known that the asymptotic invariant manifolds around an unstable periodic orbit in conservative systems can be represented by convergent series (Cherry 1926, Moser 1956, 1958, Giorgilli 2001). The unstable and stable manifolds…
Safe UAV navigation is challenging due to the complex environment structures, dynamic obstacles, and uncertainties from measurement noises and unpredictable moving obstacle behaviors. Although plenty of recent works achieve safe navigation…
The direct parametrisation method for invariant manifold is a model-order reduction technique that can be applied to nonlinear systems described by PDEs and discretised e.g. with a finite element procedure in order to derive efficient…
We develop a validated numerical procedure for continuation of local stable/unstable manifold patches attached to equilibrium solutions of ordinary differential equations. The procedure has two steps. First we compute an accurate high order…
Self-consistent chaotic transport is studied in a Hamiltonian mean-field model. The model provides a simplified description of transport in marginally stable systems including vorticity mixing in strong shear flows and electron dynamics in…
We present a continuation method that enables one to track or continue branches of periodic orbits directly in an experiment when a parameter is changed. A control-based setup in combination with Newton iterations ensures that the periodic…
Symmetry reduction by the method of slices quotients the continuous symmetries of chaotic flows by replacing the original state space by a set of charts, each covering a neighborhood of a dynamically important class of solutions,…
A compact and efficient numerical method is described for studying plane flows of an ideal fluid with a smooth free boundary over a curved and nonuniformly moving bottom. Exact equations of motion in terms of the so-called conformal…
We study the local dynamics near general unstable traveling waves of the 3D Gross-Pitaevskii equation in the energy space by constructing smooth local invariant center-stable, center-unstable and center manifolds. We also prove that (i) the…