Robustness of periodic orbits in the presence of noise

For a smooth dynamical system xn+1 = f (C, xn) (depending on a parameter C), there may be infinitely many periodic windows, that is, intervals inC having a region of stable periodic behaviour. However, the smaller of these windows are easily destroyed with tiny perturbations, so that only finitely many of the windows can be detected for a given level of noise. For a fixed perturbation size , we consider the system behaviour in the presence of noise. We look at the ‘ -robust windows’, that is, those periodic windows such that for the superstable parameter value C in that window, the general periodic behaviour persists despite noise of amplitude . We focus on the quadratic map, and numerically compute the number of periodic windows that are -robust. We obtain a robustness exponent α = 0.51 ± 0.03, which characterizes the robustness of periodic windows in the presence of noise.