Theoretical analysis for critical fluctuations of relaxation trajectory near a saddle-node bifurcation

Abstract

A Langevin equation whose deterministic part undergoes a saddle-node bifurcation is investigated theoretically. It is found that statistical properties of relaxation trajectories in this system exhibit divergent behaviors near a saddle-node bifurcation point in the weak-noise limit, while the final value of the deterministic solution changes discontinuously at the point. A systematic formulation for analyzing a path probability measure is constructed on the basis of a singular perturbation method. In this formulation, the critical nature turns out to originate from the neutrality of exiting time from a saddle-point. The theoretical calculation explains results of numerical simulations.

18pages, 17figures.The version 2, in which minor errors have been fixed, will be published in Phys. Rev. E

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