Mathematics – Dynamical Systems
Scientific paper
2004-03-16
Mathematics
Dynamical Systems
38 pages; 13 figures Keywords: sensitive dependence on initial conditions, physical measure, singular-hyperbolicity, expansive
Scientific paper
We comment on the Lorenz equations an its attractor, whose existence was rig- orously proved only around the year 2000 with a computer assisted proof together with an extension of the hyperbolic theory developed to encompass attractors robustly containing equi- libria. We present the main results of the singular-hyperbolic theory. After reviewing known results on sensitiveness and also on robustness of attractors for three- dimensional flows, together with observations on their proofs, we show that for attractors of three-dimensional flows, robust chaotic behavior is equivalent to the existence of certain hyper- bolic structures, known as singular-hyperbolicity. These structures, in turn, are associated to the existence of physical measures: in low dimensions, robust chaotic behavior for flows ensures the existence of a physical measure. We then give more details on recent results on the dynamics of singular-hyperbolic (Lorenz- like) attractors: (1) there exists an invariant foliation whose leaves are forward contracted by the flow; (2) there exists a positive Lyapunov exponent at every orbit; (3) attractors in this class have zero volume if the flow is C 2, or else the flow is globally hyperbolic; (4) there is a unique physical measure whose support is the whole attractor and which is the equilibrium state with respect to the center-unstable Jacobian; (5) the hitting time associated to a geometric Lorenz attractor satisfies a logarithm law; (6) the geometric Lorenz flow satisfies the Almost Sure Invariance Principle and the Central Limit Theorem; (7) the rate of decay of large deviations for the volume measure on the ergodic basin of a geometric Lorenz attractor is exponential; (8) the geometric Lorenz flow exhibits robust exponential decay of correlations.
Araujo Vitor
Pacifico Maria Jose
No associations
LandOfFree
Lorenz attractor and singular-hyperbolicity does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Lorenz attractor and singular-hyperbolicity, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Lorenz attractor and singular-hyperbolicity will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-574567