Spectral Analysis of Noisy Oscillators near Hopf Bifurcations ∗

We compare the dynamics of nonlinear noisy oscillators near the two types of the Hopf bifurcation. Prior to the bifurcation, in the regime of damped oscillations around the stable focus, noise serves as a bifurcation precursor: the power spectrum includes a peak at the frequency of the self-sustained oscillations. Super-and sub-critical Hopf bifurcations differ crucially in the noise dependence of the width of this spectral line. In case of a super-critical bifurcation the width is a monotonically growing function of the noise intensity. In contrast, for a sub-critical bifurcation, growth of the intensity of the weak noise enforces a decrease of the peak width; the width starts to grow only when the noise level exceeds a certain threshold value. Since the inverse spectral width is a measure of coherence, we conclude that true noise-induced coherence can be found only near the sub-critical bifurcation.