Nonhyperbolic attractors and their statistical properties

In the present talk on the basis of method proposed by Lai et al for diagnosing 2-dimensional (2-D) chaotic saddles we present a numerical procedure to distinguish hyperbolic and nonhyperbolic chaotic attractors in three-dimensional flow systems. This technique is based on calculating the angles between stable and unstable manifolds along a chaotic trajectory in R^3. We show for three-dimensional flow systems that this serves as an efficient characteristic for exploring chaotic differential systems. Further the attention is paid to the analysis of the effect of noise on nonhyperbolic chaotic attractors. It is shown that in the presence of noise certain properties of nonhyperbolic chaos can be similar to those of hyperbolic and almost hyperbolic chaos. We attempt to give numerical evidence to the existence of stationary probability measure on noisy nonhyperbolic attractors. For this purpose we apply two different methods for finding the probability distribution and then compare the obtained results in detail. The first method is based on solution of stochastic equations. The second one deals with equations for probability distiribution density.