A chaotic Hopf bifurcation in coupled maps

Three-dimensional maps with a one-dimensional invariant subspace are considered in which the dynamics in the invariant subspace is chaotic. Such maps arise from three coupled one-dimensional maps. If the coupling is uni-directional and identical, then the system has a Z(3) symmetry and this forces the two normal Lyapunov exponents to coincide. It is then possible to define a linearised average rotation about the invariant subspace, When the normal Lyapunov exponents change sign, a chaotic Hopf bifurcation occurs. By considering similar coupled systems but with different coupling strengths, the Z(3) symmetry is lost but there is still an SO(2) symmetry on the normal linearisation and so similar 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 numerical examples. (C) 1998 Elsevier Science B.V.