Symmetry breaking in reaction-diffusion systems
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(Redirected from Symmetry breaking)
Author: Dr. Dwight Barkley, Mathematics, University of Warwick, UK
Reaction-diffusion systems often possess symmetries. Specifically, the equations describing a reaction-diffusion system often are left unchanged by certain groups of transformations, such as reflection, translation or rotation. Symmetry breaking refers to the situation in which solutions to the equations have less symmetry than the equations themselves.
For example, in the simplest case a reaction-diffusion system might be left-right symmetric as in figure 1. States of the system may fail to be left-right symmetric. Such broken-symmetry states will normally come in pairs, related to one other by reflection.
Symmetric breaking is important in the study of reaction-diffusion equations because often the solutions arise via symmetry breaking bifurcations and behavior of solutions is strongly dictated by the nature of the symmetry breaking. Many features of symmetry breaking follow from considerations of equivariant bifurcation theory and equivariant dynamical systems, and are independent of the details of the underlying reaction-diffusion system.
It is frequently observed that in systems with several symmetries, not all symmetries break at once as a control parameter of the system is varied. For example, in a system with both translation and reflection symmetry, often one observes that as a control parameter is varied translation symmetry breaks first. Further variation in the control parameter lead to breaking of reflection symmetry.
Reaction-Diffusion Equations in One Dimension
Reaction-diffusion Equations in Multiple Dimensions
Many examples possible. The most interesting perhaps in Euclidian symmetry breaking.
