Emergence of Effective Low-Dimensional Dynamics in the Macroscopic Behaviour of Coupled Map Lattices

Bifurcation of democratically coupled logistic maps in various space dimensions are studied beyond the accumulation point of the direct cascade of the individual maps. In dimensions d = 2 and 3, only subharmonic bifurcations between periodic collective states are observed upon increasing the control parameter. The case d = 4 displays more complicated sequences with subcritical bifurcations and attractor coexistence. In dimension five or more, even more nontrivial behaviours become possible. An example of quasi-periodic collective motion is given for d = 5. The general implications of these preliminary results are discussed.