Multi-affinity and multi-fractality in systems of chaotic elements with long-wave forcing

Abstract

Multi-scaling properties in quasi-continuous arrays of chaotic maps driven by long-wave random force are studied. The spatial pattern of the amplitude X(x, t) is characterized by multi-affinity, while the field defined by its coarse-grained spatial derivative Y(x, t) := |(X(x + δ, t) − X(x, t))/δ| exhibits multi-fractality. The strong behavioral similarity of the X- and Y-fields respectively to the velocity and energy dissipation fields in fully-developed fluid turbulence is remarkable, still our system is unique in that the scaling exponents are parameter-dependent and exhibit nontrivial q-phase transitions. A theory based on a random multiplicative process is developed to explain the multi-affinity of the X-field, and some attempts are made towards the understanding of the multi-fractality of the Y-field.