Transitivity and blowout bifurcations in a class of globally coupled maps

A class of globally coupled one dimensional maps is studied. For the uncoup led one dimensional map it is possible to compute the spectrum of Liapunov exponents exactly, and there is a natural equilibrium measure (Sinai-Ruelle -Bowen measure), so the corresponding 'typical' Liapunov exponent may also be computed. The globally coupled systems thus provide examples of blowout bifurcations in arbitrary dimension. In the two dimensional case these maps have parameter values at which then is a transitive (topological) attracto r which is a filled-in quadrilateral and, simultaneously, the synchronized state is a Milnor attractor. (C) 1999 Elsevier Science B.V. All rights rese rved.