Greedy Kernel Methods for Center Manifold Approximation

TL;DR

This paper introduces a data-driven greedy kernel approach to approximate center manifolds in dynamical systems, enabling simplified stability analysis near non-hyperbolic equilibria with promising accuracy.

Contribution

It presents a novel kernel-based greedy method for approximating center manifolds from simulation data, improving qualitative analysis of complex dynamical systems.

Findings

Method shows promising performance on various examples

Achieves good accuracy in manifold approximation

Enables simplified stability analysis near equilibria

Abstract

For certain dynamical systems 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 a non-hyperbolic equilibrium point, and to obtain meaningful predictions of its behavior by analyzing a reduced dimensional problem. Since the manifold is usually not known, approximation methods are of great interest to obtain qualitative estimates. In this work, we use a data-based greedy kernel method to construct a suitable approximation of the manifold close to the equilibrium. 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 to construct a surrogate model of the manifold. The method is tested on different examples which show promising performance and good accuracy.