Periodicity manifestations in the turbulent regime of the globally coupled map lattice

Abstract

We revisit the globally coupled map lattice (GCML). We show that in the so called turbulent regime various periodic cluster attractor states are formed even though the coupling between the maps are very small relative to the nonlinearity in the element maps. Most outstanding is a maximally symmetric three cluster attractor in period three motion (MSCA) due to the foliation of the period three window of the element logistic maps. An analytic approach is proposed which explains successfully the systematics of various periodicity manifestations in the turbulent regime. The linear stability of the period three cluster attractors is investigated. 05.45.+b,05.90.+m

Universality in globally coupled maps and flows

“…The other is a self-organization of a certain attractor by synchronization [2] in a complex system under conflict between randomness and coherence. One impressive model embodying this phenomenon is Globally Coupled Map Lattice (GCML) [3], [4]. In this paper we study Globally Coupled Flow Lattice (GCFL) [5] and show that the universality at the level of the constituents may be extended to the level of the whole system, that is, our GCFL has much in common with GCML.…”

“…Here we consider its simplest form, where the map is a logistic map with the same a and the coupling is also common (homogeneous GCML) [3] [4]. It evolves in an iteration of two steps.…”

Periodicity manifestations in the non-locally coupled maps

“…where W P Q is the couplings in (5), (6), (7). If the spatial correlation between the maps are negligible, the variance of ξ P may be estimated at each time t as…”

“…The most prominent PM's are induced by the period three window [6]. If GCML is on the foliation curves of the p3 window, the maps organize themselves into almost equally populated three clusters, which oscillate mutually in period three -p3c3 maximally symmetric cluster attractor (MSCA) [7]. With slightly higher ε at the same a, that is, on the foliation curves from the intermittent region, the maps organize themselves in p3c2 cluster attractor.…”

“…This was called as a hidden coherence and triggered much scrutiny [3][4][5]. Furthermore, it has been recently found that the periodic windows of the element map systematically foliate and produce various periodicity manifestations (PM's) [6][7][8][9]. Therefore, GCML in the turbulent regime is a system which sensitively mirrors the periodic windows with the background hidden coherence.…”