Critical fluctuations near the pitchfork bifurcations of period-doubling maps

Period-doubling maps, such as the logistic map, have been a subject of intense study in both physics and biology.~ The period-doubling route to chaos proceeds through a sequence of supercritical pitchfork bifurcations.~ Here, motivated by applications to population ecology, we investigate the asymptotic behavior of period-doubling bifurcations subject to environmental or demographic noise.~ We demonstrate, analytically, that fluctuations in the vicinity of each noisy pitchfork bifurcation are described by finite-size mean-field theory.~ Our results establish an exact correspondence between the bifurcations of far-from-equilibrium systems and the mean-field critical phenomena of equilibrium systems.