Capturing the Global Behavior of Dynamical Systems with Conley-Morse Graphs

  • Zin AraiEmail author
  • Hiroshi Kokubu
  • Ippei Obayashi
Conference paper


We present a computational machinery for describing and capturing the global qualitative behavior of dynamical systems (Arai et al. SIAM J Appl Dyn Syst 8:757–789, 2009). Given a dynamical system, by subdividing the phase space into a finite number of blocks, we construct a directed graph which represents the topological behavior of the system. Then we apply fast graph algorithms for the automatic analysis of the dynamics. In particular, the dynamics can be easily decomposed into recurrent and gradient-like parts which allows further analysis of asymptotic dynamics. The automatization of this process allows one to scan large sets of parameters of a given dynamical system to determine changes in dynamics automatically and to search for “interesting” regions of parameters worth further attention. We also discuss an application of the method to time series analysis. The method presented in Sects. 1 –4 below is given in [1] for the first time, which is based on and combines a number of theoretical results as well as computational software packages developed earlier. For the details, see the original paper [1].


  1. 1.
    Arai, Z., Kalies, W., Kokubu, H., Mischaikow, K., Oka, H. and Pilarczyk, P., SIAM Journal on Applied Dynamical Systems, 8 (2009), 757–789.CrossRefGoogle Scholar
  2. 2.
    Arai, Z., Kokubu, H. and Pilarczyk, P., Japan Journal of Industrial and Applied Mathematics,26 (2009) 393–417.Google Scholar
  3. 3.
    Conley, C., Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series in Math., 38, Amer. Math. Soc., Providence, RI, 1978.Google Scholar
  4. 4.
    Mischaikow, M. and Mrozek, M., Conley index, Handbook of dynamical systems, Vol. 2, North-Holland (2002), 393–460.Google Scholar
  5. 5.
    Ugarcovici, I. and Weiss, H., Nonlinearity 17 (2004) 1689–1711.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Creative Research InstituteHokkaido UniversitySapporoJapan
  2. 2.JST CRESTTokyoJapan
  3. 3.Department of MathematicsKyoto UniversityKyotoJapan

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