The disordered logistic map

Abstract

The logistic map is a quintessential model in the study of low-dimensional chaos. High-dimensional chaos, on the other hand, presents itself in disordered systems with many interacting and heterogeneously coupled components. Here, we formulate a system of many logistic maps, interacting through disordered couplings. Using a combination of dynamic mean-field theory, random matrix theory and numerical simulations, we show that even the smallest amount of disorder can remove the period-doubling cascade in the conventional logistic map. Instead we find a transition to high-dimensional chaos, marked by an oscillatory instability not previously reported for disordered systems. We also show that with sufficiently strong homogeneous coupling between the maps one recovers elements of the conventional period-doubling cascade. Our findings indicate that well-known phenomena in dynamical systems can be fragile in the face of disorder. At the same time, new phenomena emerge that are neither found in simple low-dimensional dynamics nor in high-dimensional disordered systems.