Effects of microscopic disorder on the collective dynamics of globally coupled maps

Abstract

This paper studies the effect of independent additive noise on the synchronous dynamics of large populations of globally coupled maps. Our analysis is complementary to the approach taken by Teramae and Kuramoto [39] who pointed out the anomalous scaling properties preceding the loss of coherence. We focus on the macroscopic dynamics that remains deterministic at any noise level and differs from the microscopic one. Using properly defined order parameters, an analytical approach is proposed for describing the collective dynamics in terms of an approximate lowdimensional system. The systematic derivation of the macroscopic equations provides a link between the microscopic features of the population (single-element dynamics and noise distribution) and the properties of the emergent behaviour. The macroscopic bifurcations induced by noise are compared to those originating from parameter mismatches within the population.

Integrable order parameter dynamics of globally coupled oscillators

“…When the coupling is absent (K = 0) system (1) describes the dynamics of noninteracting oscillators with a stable circular limit cycles |z j | = 1. In [10] system (1) was studied in the framework of two order parameter approximation (for more about the order parameter expansion method, see [11][12][13][14][15][16]):…”

Phase transitions induced by microscopic disorder: A study based on the order parameter expansion

“…From the practical point of view, the models we will be considering bear some similarities with random-field, or impurities, models. As tool of investigation we will refine a previously developed order parameter expansion method of approximating large systems of coupled differential equations [25,26,27,28,29] with diverse parameters. This allows the reduction of the large set of differential equations to just three: one for the global, mean field, value and two which describe the fluctuations around this mean value.…”

Low-Dimensional Chaos in Populations of Strongly-Coupled Noisy Maps

“…For instance, as shown by Kuramoto and Teramae, even weak noise can cause the population distribution moments to scale anomalously with respect to the noise distribution moments over a range of scales. 11), 15) In recent work, we have focused on how the addition of noise to a synchronous regime modifies the collective dynamics. 12), 15) Somewhat counterintuitively, although noise blurs the trajectory of any population element, it affects the mean field of infinite populations in a deterministic way.…”

“…where A q and Γ i depend on the first 2P moments of the noise distribution. 15) Of the remaining order parameters, those up to q = n P (P being the degree of the the map f ) have a dynamics slaved to the the first n, while the others are constantly equal to the noise distribution moment of corresponding order.…”

“…This paper follows a series of previous works in which, in particular, we have introduced a systematic order parameter expansion which approximates with increasing accuracy the macroscopic dynamics observed at strong coupling values by typeset using PTPT E X.cls Ver.0.9 low-dimensional effective maps for a small set of macroscopic observables. 13), 14), 15) Here we use this expansion scheme to tackle the issue of the actual dimensionality of the collective chaos observed.…”