Reaction—Diffusion Systems and Interface Dynamics

  • Hazime Mori
  • Yoshiki Kuramoto


Existing studies on reaction—diffusion systems have been carried out with the Belousov-Zhabotinskii reaction acting as the standard example. Approaches based on amplitude equations are in general not suited to describe patterns peculiar to such ‘excitable’ systems because, while excitability originates in a particular property of global flow in phase space, amplitude equations are obtained by considering only local flow. In fact, BZ reaction systems display wave patterns that lack both the temporal and spatial smoothness displayed by solutions to amplitude equations, and thus it is much more natural to treat excitable systems using local interface structure representing the sudden change in state of a system over a small distance. Patterns containing interfaces arise out of the cooperative dynamics of degrees of freedom undergoing rapid and slow temporal change. Rapidly changing degrees of freedom form a bistable partial subsystem consisting of two states, an active (or excitable) state and an inactive (or nonexcitable) state. The speed with which the transition between these two states is carried out is controlled by the slowly changing degrees of freedom.


Propagation Speed Spiral Wave Phase Field Model Amplitude Equation Excitable System 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Kobayashi, R. (1993) Modeling and numerical simulations of dendritic crystal growth. Physica D 63, 410–423ADSMATHCrossRefGoogle Scholar
  2. Meron, E. (1992) Pattern formation in excitable media. Phys. Rep. 218, 1–66MathSciNetADSCrossRefGoogle Scholar
  3. Ouyang, Q., Swinney, H.L. (1991) Transition from uniform state to hexagonal and striped Turing patterns. Nature 352, 610–612ADSCrossRefGoogle Scholar
  4. Pelcé, P., Sun, J. (1991) Wave front interaction in steadily rotating spirals. Physica D 48, 353–366ADSMATHCrossRefGoogle Scholar
  5. Rinzel, J., Keller, J.B. (1973) Travelling wave solutions of a nerve conduction equation. Biophys. J. 48, 1313–1337CrossRefGoogle Scholar
  6. Skinner, G.S., Swinney, H. (1991) Periodic to quasiperiodic transition of chemical spiral rotation. Physica D 48, 1–16ADSMATHCrossRefGoogle Scholar
  7. Winfree, A.T. (1974) Sci. Am. 230 No. 6, 82CrossRefGoogle Scholar
  8. Winfree, A.T. (1980) The Geometry of Biological Time. Springer, Berlin, HeidelbergGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Hazime Mori
    • 1
    • 2
  • Yoshiki Kuramoto
    • 3
  1. 1.Kyushu UniversityJapan
  2. 2.Higashi-ku FukuokaJapan
  3. 3.Graduate School of SciencesKyoto UniversityKyotoJapan

Personalised recommendations