Virtual Hopf phenomenon: A new precursor of period-doubling bifurcations

A new type of noise-induced precursor of a period-doubling bifurcation is described. The usual noise rise in the power spectrum at half the fundamental frequency, typical just before the onset of period doubling, may be preceded by a pair of symmetrically located noise bumps characteristic of a Hopf bifurcation precursor. The continuous evolution from the Hopf precursor to the period-doubling precursor may be understood in terms of the behavior of the Floquet multipliers as a control parameter is varied. The necessary appearance of the virtual Hopf phenomenon may be proven for a large class of low-dimensional systems, including the driven damped pendulum and the Lorenz equations, between each successive period doubling in the infinite cascade leading to chaos.