We consider complex dynamical systems showing metastable behavior but nolocal separation of fast and slow time scales. The article raises the questionof whether such systems exhibit a low-dimensional manifold supporting itseffective dynamics. For answering this question, we aim at finding nonlinearcoordinates, called reaction coordinates, such that the projection of thedynamics onto these coordinates preserves the dominant time scales of thedynamics. We show that, based on a specific reducibility property, theexistence of good low-dimensional reaction coordinates preserving the dominanttime scales is guaranteed. Based on this theoretical framework, we develop andtest a novel numerical approach for computing good reaction coordinates. Theproposed algorithmic approach is fully local and thus not prone to the curse ofdimension with respect to the state space of the dynamics. Hence, it is apromising method for data-based model reduction of complex dynamical systemssuch as molecular dynamics.
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