Emergence of quasiperiodicity in symmetrically coupled, identical period-doubling systems

When two Identical period-doubling systems are coupled symmetrically, the period-doubling transition to chaos may be replaced by a quasiperiodic transition. The reason for this is that at an early stage of the period-doubling cascade, a Hopf bifurcation instead of a period-doubling bifurcation occurs. Our main result is that the emergence of this Hopf bifurcation is a generic phenomenon in symmetrically coupled, identical period-doubling systems. The whole phenomenon is stable against small nonsymmetric perturbations. Our results cover maps and differential equations of arbitrary dimension. As a consequence the Feigenbaum transition to chaos in these coupled systems-which exists, but tends to be unstable-is accompanied by an infinity of Hopf bifurcations.