Critical exponents near a bifurcation point in noise-induced escape over a fluctuating barrier

If a bistable system is brought closer to the bifurcation point at A= A(c) , the rate of noise-activated escape to the other stable equilibrium increases as ln k proportional, variant- mid R:A- A(c) mid R:(3/2) . Recently, Dykman, Golding, and Ryvkine derived that upon the addition of an oscillating force, the exponent equals 2 in an intermediate-frequency regime [Phys. Rev. Lett. 92, 080602 (2004)]; Phys. Rev. E 69, 061102 (2004)]]. In this paper we study the critical exponents near a bifurcation point for an analytically solvable case; that of a piecewise linear well with an imposed dichotomous fluctuation. It is shown how an exponent 2 emerges when a parameter is changed to move a system from nonadiabaticity to adiabaticity. It is, furthermore, shown that the power law breaks down for low barriers. Finally, it is explained why fundamental differences occur between smooth and piecewise linear setups.