Symmetric patterns in linear arrays of coupled cells

Chaos
Irving R Epstein, Martin Golubitsky

Abstract

In this note we show how to find patterned solutions in linear arrays of coupled cells. The solutions are found by embedding the system in a circular array with twice the number of cells. The individual cells have a unique steady state, so that the patterned solutions represent a discrete analog of Turing structures in continuous media. We then use the symmetry of the circular array (and bifurcation from an invariant equilibrium) to identify symmetric solutions of the circular array that restrict to solutions of the original linear array. We apply these abstract results to a system of coupled Brusselators to prove that patterned solutions exist. In addition, we show, in certain instances, that these patterned solutions can be found by numerical integration and hence are presumably asymptotically stable.

References

Jun 11, 1990·Physical Review Letters·V Castets De Kepper P

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Citations

Apr 8, 2010·Chaos·Huifeng ZhuGemunu H Gunaratne
Jul 13, 2004·Physical Review Letters·Atsuko TakamatsuTeruo Fujii
Dec 21, 2011·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Giovanni Russo, Jean-Jacques E Slotine
Dec 20, 2003·Physical Review Letters·Visarath InBrian K Meadows
Nov 13, 2012·RSC Advances·Yevgeniy V KalininDavid H Gracias
Sep 1, 1994·Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics·J F HeagyL M Pecora
Feb 9, 2021·PLoS Computational Biology·Rocky DiegmillerKonstantin Mischaikow
Feb 9, 2005·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Wenyuan LiuJunzhong Yang
Jul 9, 1993·Science·J E Pearson

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