Replica-symmetry breaking in dynamical glasses

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Systems of globally coupled logistic maps (GCLM) can display complex collective behaviour characterized by the formation of synchronous clusters. In the dynamical clustering regime, such systems possess a large number of coexisting attractors and might be viewed as dynamical glasses. Glass properties of GCLM in the thermodynamical limit of large system sizes N are investigated. Replicas, representing orbits that start from various initial conditions, are introduced and distributions of their overlaps are numerically determined. We show that for fixed-field ensembles of initial conditions, as used in previous numerical studies, all attractors of the system become identical in the thermodynamical limit up to variations of order 1/√N because the initial value of the coupling field is characterized by vanishing fluctuations, and thus replica symmetry is recovered for N→ ∞. In contrast to this, when random-field ensembles of initial conditions are chosen, replica symmetry remains broken in the thermodynamical limit.