Influence of random fluctuations on delayed bifurcations. II. The cases of white and colored additive and multiplicative noise

The influence of noise on the delay of a bifurcation point in the presence of a swept control parameter has been investigated theoretically, by digital simulation and by analog electronic experiment. The results obtained in an earlier paper [N. G. Stocks, R. Mannella, and P. V. E. McClintock, Phys. Rev. A 40, 5361 (1989)] have thereby been extended and complemented. In particular, exact analytic expressions have been derived for the time-dependent probability densities P(x,t), and these have been used to obtain the mean first-passage time ${\mathit{t}}_{\mathrm{MFPT}}^{\mathrm{*}}$ for ${\mathit{x}}^2$(t) to reach a threshold under the influence of Gaussian fluctuations, in several contexts: additive external white noise, additive external exponentially correlated noise, additive internal white noise, additive internal exponentially correlated noise, multiplicative white and colored noise. Based on Zeghlache, Mandel, and Van den Broeck’s [Phys. Rev. A 40, 286 (1989)] alternative definition of the bifurcation time ${\mathit{t}}_{\mathrm{moment}}^{\mathrm{*}}$ in terms of the evolution of the second moment 〈${\mathit{x}}^2$(t)〉, an expression is derived for ${\mathit{t}}_{\mathrm{moment}}^{\mathrm{*}}$ for the general case of combined additive and multiplicative noises. The calculations are tested by comparison with the results of analog experiments and digital simulations, with which they are shown to be in excellent agreement.