Effect of Noise on Nonhyperbolic Chaotic Attractors

We consider the effect of small noise of maximum amplitude $\varepsilon$ on a chaotic system whose noiseless trajectories limit on a fractal strange attractor. For the case of nonhyperbolic attractors of two-dimensional maps the effect of noise can be much stronger than for hyperbolic attractors. In particular, the maximum over all noisy orbit points of the distance between the noisy orbit and the noiseless nonhyperbolic attractor scales like ${\varepsilon }^{{1/D}_1}$ ( $D_1>1\mathrm{}$ is the information dimension of the attractor), rather than like $\varepsilon$ (the hyperbolic case). We also find a phase transition in the scaling of the time averaged moments of the deviations of a noisy orbit from the noiseless attractor.