Homoclinic tangencies and non-normal Jacobians - Effects of noise in nonhyperbolic chaotic systems

A general study of the effects of noise interacting with the deterministic dynamics of nonhyperbolic chaotic 2-D maps and 3-flows is presented. Noise in these systems can have dramatic effects on the invariant measure of the system. In regions of homoclinic tangencies perturbations are transported away from the neighborhood of the attractor, leading to deformations which can be one to two orders of magnitude larger than the noise level. A qualitative understanding is attained by a study of the universal properties of the unstable and stable foliation of the phase space in the vicinity of homoclinic tangencies. Through the investigation of nontrivial structures of the tangent space at homoclinic tangencies related to the nonnormal Jacobians we obtain a quantitative description of these noise-induced deformations of the attractor. Local expansion rates which are much larger than the maximal Lyapunov exponent can be used as a measure of the system's structural instability under perturbations. We exemplify our general results on the Henon, Ikeda and Duffing system. A new and effective algorithm to calculate the homoclinic tangencies in the entire phase space based on the results of this paper is presented.