Noise-induced escape from attractors in one-dimensional maps

The addition of external noise to a dynamical system described by an iterated map causes the orbit to escape from the attractor. The escape time τ has the behavior τ≊${\tau }_0$exp($E_0$/Γ), where Γ is the noise temperature, $E_0$ is the minimum escape energy, and ${\tau }_0$ is the inverse of the attempt rate. We will describe an analytical method for calculating the mean escape time based on the principle of minimum escape energy. Analytical solutions for $E_0$ are presented for values of the mapping control parameter a close to tangent bifurcations and interior crises. The minimum escape energy displays a power-law dependence on the control parameter near tangent bifurcations ($E_0$∼‖a-$a_t$ ${\Vert }^{3/2}$) and near interior crises ($E_0$∼‖a-$a_c$ ${\Vert }^2$). Numerical solutions are given for control-parameter values throughout the range of the attractor. The results agree with the results of Monte Carlo simulations of the logistic map and with independent work on the noise stability of rf-driven Josephson junctions.