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

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