Traveling and Stationary Patterns in Bistable Reaction-Diffusion Systems on Network

  • Nikos E. Kouvaris
  • Hiroshi Kori
  • Alexander S. Mikhailov
Conference paper
Part of the Springer Proceedings in Complexity book series (SPCOM)


Traveling and stationary patterns in bistable reaction-diffusion systems have been extensively studied for classical continuous media and regular lattices. Here, we consider analogs of such non-equilibrium patterns in bistable one-component systems on trees and on random networks. As revealed through numerical simulations, traveling fronts exist in network-organized systems. They represent waves of transition from one stable state into another, spreading over the entire network. The fronts can furthermore be pinned, thus forming stationary structures. While pinning of fronts has previously been considered for chains of diffusively coupled bistable elements, the network architecture brings about significant differences. An important role is played by the degree (the number of connections) of a node. For regular trees with a fixed branching factor, the pinning conditions are analytically determined. For large random networks, the mean-field theory for stationary patterns is constructed.


Self-organization Pattern formation Nonlinear dynamics Bistability Complex networks Traveling fronts Pinning Stationary patterns 



Financial support from the DFG Collaborative Research Center SFB910 “Control of Self-Organizing Nonlinear Systems” and from the Volkswagen Foundation in Germany is gratefully acknowledged.


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Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Nikos E. Kouvaris
    • 1
  • Hiroshi Kori
    • 2
  • Alexander S. Mikhailov
    • 1
  1. 1.Department of Physical ChemistryFritz Haber Institute of the Max Planck SocietyBerlinGermany
  2. 2.Department of Information SciencesOchanomizu UniversityTokyoJapan

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