Phase ordering at crises

Phase order, namely the average direction of sequential iterations, is studied in the family of unimodal maps x --> 1 - mu\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 similar to (mu - mu(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 rho(,V) at the edges of the interval: as x --> +/- 1, rho(x) similar to (1 - \x\(z))(-beta) with beta = 1 - 1/z. (C) 2002 Elsevier Science B.V. All rights reserved.