Phase ordering at crises

Abstract

Phase order, namely the average direction of sequential iterations, is studied in the family of unimodal maps x→1≄μ|x|^z on the interval [≄1,1]. The average phase order or "magnetization" M is sensitive to local changes in the dynamics. At merging crises, this quantity increases from zero with the scaling behaviour M ^∼(μ≄μ_c)^1/z, while at exterior crises, M decreases, also having the same scaling exponent. We find that the exponent z is governed by the singularities of the invariant density ρ(x) at the edges of the interval: as x→±1, ρ(x)^∼(1≄|x|^z)^-βwith β=1≄1/z.