Pattern formation and period doublings in the many-body coupled logistic maps

Abstract

In this work we consider two many-body generalizations of the logistic map, where we couple single maps with nearest-neighbour interactions on a one-dimensional lattice, getting a discrete-time nonlinear dynamical system. Numerically looking at some period $2^k$-tupling order parameters, we check that at least for $k\in\{1,\,2,\,3,\,4\}$ the period-doubling transitions of the single map persist. The values of the driving parameter $\mu$ where they occur remain unchanged, independently of the system size, and of the type and the strength of the coupling in the many-body system. We numerically observe that the nonlinear dynamics leads to the formation of patterns, that can occur if $\mu$ lies beyond the first period-doubling threshold $\mu_2=3$ and appear if the system is large enough to accommodate them. We study the properties of the patterns -- the characteristic length scale and the amplitude -- and find that the former changes over many orders of magnitude when $\mu$ is varied. If the system size is large enough, the properties of the pattern have no relation at all with the single logistic map, witnessing the qualitative difference of the many-body dynamics from the single-map one. We discuss also the effect of noise and the relation of our findings with time crystals.Comment: 9 pages, 3 figure