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Congestion and Almost Invariant Sets in Dynamical Systems

  • Michael Dellnitz
  • Robert Preis
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2630)

Abstract

An almost invariant set of a dynamical system is a subset of state space where typical trajectories stay for a long period of time before they enter other parts of state space. These sets are an important characteristic for analyzing the macroscopic behavior of a given dynamical system. For instance, recently the identification of almost invariant sets has successfully been used in the context of the approximation of so-called chemical conformations for molecules.

In this paper we propose new numerical and algorithmic tools for the identification of the number and the location of almost invariant sets in state space. These techniques are based on the use of set oriented numerical methods by which a graph is created which models the underlying dynamical behavior. In a second step graph theoretic methods are utilized in order to both identify the number of almost invariant sets and for an approximation of these sets. These algorithmic methods make use of the notion of congestion which is a quantity specifying bottlenecks in the graph. We apply these new techniques to the analysis of the dynamics of the molecules Pentane and Hexane. Our computational results are compared to analytical bounds which again are based on the congestion but also on spectral information on the transition matrix for the underlying graph.

Keywords

Cost Function Short Path Invariant Measure External Cost Graph Partitioning 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Michael Dellnitz
    • 1
  • Robert Preis
    • 1
    • 2
  1. 1.Department of Mathematics and Computer ScienceUniversität PaderbornPaderbornGermany
  2. 2.Computer Science Research InstituteSandia National LaboratoriesAlbuquerqueUSA

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