Imperfect pitchfork bifurcation with weak noise and an application to superfluid turbulence in liquid helium

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Abstract
In this paper we analyze a model containing the unfolded normal form of the imperfect pitchfork bifurcation with additional terms representing additive white noise and a linear Ornstein-Uhlenbeck noise. After linearizing the model about a branch of stable steady states, we investigate the variance of fluctuations about these steady states as a function of a control parameter. We apply our model to experimental data obtained by Griswold, Lorensen, and Tough [Phys. Rev. B 35, 3149 (1987)] on the transition between superfluid turbulent states T-I and T-II in liquid-helium counterflow experiments. Our model accounts qualitatively for the steady states, relaxation times, and the variance of fluctuations which were measured in this system. Furthermore, our model predicts the existence of a curve of unstable steady states which have not yet been observed. Perturbations from the known steady state in the direction of these new unstable steady states are expected to lead to qualitatively new dynamics.

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