A CHAOTIC HOPF-BIFURCATION IN COUPLED MAPS

Three-dimensional maps with a one-dimensional invariant subspace are c onsidered in which the dynamics in the invariant subspace is chaotic. Such maps arise from three coupled one-dimensional maps. If the coupli ng is uni-directional and identical, then the system has a Z(3) symmet ry and this forces the two normal Lyapunov exponents to coincide. It i s then possible to define a linearised average rotation about the inva riant subspace, When the normal Lyapunov exponents change sign, a chao tic Hopf bifurcation occurs. By considering similar coupled systems bu t with different coupling strengths, the Z(3) symmetry is lost but the re is still an SO(2) symmetry on the normal linearisation and so simil ar results for the Lyapunov exponents hold. If there is no symmetry on the linearisation either, then the multiple Lyapunov exponents split, although it is still possible to define a linearised average rotation in many cases. These three different scenarios are illustrated with n umerical examples. (C) 1998 Elsevier Science B.V.