Transition Manifolds of Complex Metastable Systems: Theory and Data-Driven Computation of Effective Dynamics

Andreas Bittracher, Péter Koltai, Stefan Klus*, Ralf Banisch, Michael Dellnitz, Christof Schütte

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

32 Citations (Scopus)
8 Downloads (Pure)


We consider complex dynamical systems showing metastable behavior, but no local separation of fast and slow time scales. The article raises the question of whether such systems exhibit a low-dimensional manifold supporting its effective dynamics. For answering this question, we aim at finding nonlinear coordinates, called reaction coordinates, such that the projection of the dynamics onto these coordinates preserves the dominant time scales of the dynamics. We show that, based on a specific reducibility property, the existence of good low-dimensional reaction coordinates preserving the dominant time scales is guaranteed. Based on this theoretical framework, we develop and test a novel numerical approach for computing good reaction coordinates. The proposed algorithmic approach is fully local and thus not prone to the curse of dimension with respect to the state space of the dynamics. Hence, it is a promising method for data-based model reduction of complex dynamical systems such as molecular dynamics.

Original languageEnglish
Pages (from-to)471-512
Number of pages42
JournalJournal of Nonlinear Science
Issue number2
Publication statusPublished - Apr 2018


  • Coarse graining
  • Effective dynamics
  • Metastability
  • Reaction coordinate
  • Transfer operator
  • Whitney embedding theorem

ASJC Scopus subject areas

  • Modelling and Simulation
  • Engineering(all)
  • Applied Mathematics


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