Zero Lyapunov exponent in the vicinity of the saddle-node bifurcation point in the presence of noise.

We consider a behavior of the zero Lyapunov exponent in the vicinity of the bifurcation point that occurs as the result of the interplay between dynamical mechanisms and random dynamics. We analytically deduce the laws for the dependence of this Lyapunov exponent on the control parameter both above and below the bifurcation point. The developed theory is applicable both to the systems with the random force and to the deterministic chaotic oscillators. We find an excellent agreement between the theoretical predictions and the data obtained by means of numerical calculations. We also discuss how the revealed regularities are expected to take place in other relevant physical circumstances.