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We prove for an infinite array of globally coupled overdamped anharmonicoscillators subject to additive Gaussian white noise the existence of awell-behaved critical manifold in the parameter space which separates asymmetric phase from a symmetry broken phase. Given two of the systemparameters there is an unique critical value of the third. The proof exploitsthat the critical control parameter a_c is bounded by its limit values for weakand for strong noise. In these limits the mechanism of symmetry breakingdiffers. For weak noise the distribution is Gaussian and the symmetry is brokenas the whole distribution is shifted in either the positive or the negativedirection. For strong noise there is a symmetric double-peak distribution andthe symmetry is broken as the weights of the peaks become different. We derivean ordinary differential equation whose solution describes the criticalmanifold. Using a series ansatz to solve this differential equation wedetermine the critical manifold for weak and for strong noise and compare it tonumerical results. We derive analytic expressions for the order parameter andthe susceptibility close to the critical manifold.

Critical manifold of globally coupled overdamped anharmonic oscillators driven by additive Gaussian white noise