Kernel methods for center manifold approximation and a data-based version of the Center Manifold Theorem

TL;DR

This paper introduces a data-driven kernel method to approximate the center manifold in dynamical systems, providing a practical approach with error quantification and testing on various examples.

Contribution

It develops a novel data-based version of the center manifold theorem using kernel methods, enabling approximation without knowing the exact manifold.

Findings

Effective approximation of the center manifold using data-based kernel methods

Quantified error bounds between approximate and true reduced dynamics

Successful testing on multiple dynamical system examples

Abstract

For dynamical systems with a non hyperbolic equilibrium, it is possible to significantly simplify the study of stability by means of the center manifold theory. This theory allows to isolate the complicated asymptotic behavior of the system close to the equilibrium point and to obtain meaningful predictions of its behavior by analyzing a reduced order system on the so-called center manifold. Since the center manifold is usually not known, good approximation methods are important as the center manifold theorem states that the stability properties of the origin of the reduced order system are the same as those of the origin of the full order system. In this work, we establish a data-based version of the center manifold theorem that works by considering an approximation in place of an exact manifold. Also the error between the approximated and the original reduced dynamics are quantified. We then use an apposite data-based kernel method to construct a suitable approximation of the manifold close to the equilibrium, which is compatible with our general error theory. The data are collected by repeated numerical simulation of the full system by means of a high-accuracy solver, which generates sets of discrete trajectories that are then used as a training set. The method is tested on different examples which show promising performance and good accuracy.