Coupled maps: An approach to spatiotemporal chaos

Abstract

The transition regime from periodic and quasiperiodic motion to spatiotemporal chaos is examined for coupled-map lattices. For periodic states the stability criteria for homogeneous solutions are determined, and the formation of checkerboard patterns is treated analytically. Also, the period-doubling route to spatiotemporal chaos is discussed. For the quasiperiodic transition, renormalization-group analysis is carried out for both forward coupling and the more generic linear coupling. This leads to scaling results for the spatiotemporal intermittent regime. In particular, a coherence length is identified, based on the distributions of the phase fluctuations and their derivatives. Finally, percolation methods in the study of spatiotemporal intermittency are numerically tested. It is shown that finite-size effects are substantial.