  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.
