# Scaling Laws and Bifurcation

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## Abstract

Equations with symmetry often have solution branches which are related by a simple rescaling. This property can be expressed in terms of a *scaling law* which is similar to the equivariance condition except that it also involves the parameters of the problem. We derive a natural context for the existence of such scaling laws based on the symmetry of the problem and show how bifurcation points can also be related by a scaling. This leads in some cases, to a proof of existence of bifurcating branches at a mode interaction. The results are illustrated for the Kuramoto-Sivashinsky equation.

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© Springer-Verlag 1991