Singular-hyperbolic attractors are chaotic
By Vitor Araujo, Maria José Pacifico, E. Pujals, Marcelo Viana. 2005
147 citations recorded by the source corpus.
We prove that a singular-hyperbolic attractor of a 3-dimensional flow is chaotic, in two different strong senses. First, the flow is expansive: if two points remain close at all times, possibly with time reparametrization, then their orbits coincide. Second, there exists a physical (or Sinai-Ruelle-Bowen) measure supported on the attractor whose ergodic basin covers a full Lebesgue (volume) measure subset of the topological basin of attraction. Moreover this measure has absolutely continuous conditional measures along the center-unstable direction, is a u-Gibbs state and is an equilibrium state for the logarithm of the Jacobian of the time one map of the flow along the strong-unstable direction. This extends to the class of singular-hyperbolic attractors the main elements of the ergodic theory of uniformly hyperbolic (or Axiom A) attractors for flows. In particular these results can be applied (i) to the flow defined by the Lorenz equations, (ii) to the geometric Lorenz flows, (iii) to the attractors appearing in the unfolding of certain resonant double homoclinic loops, (iv) in the unfolding of certain singular cycles and (v) in some geometrical models which are singular-hyperbolic but of a different topological type from the geometric Lorenz models. In all these cases the results show that these attractors are expansive and have physical measures which are u-Gibbs states.
Subfields: Control and Systems Engineering, Mathematical Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/3352f5f640a348cac31b3a2a3a6940cc0eba3a47
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Control and Systems Engineering
In Mathematical Physics
In Statistical and Nonlinear Physics
Citations in the corpus
- Stochastic climate dynamics: Random attractors and time-dependent invariant measures (2011, background): “This result has been extended recently to the Lorenz flow [3] itself, in which the SRB measure is supported by a strange attractor of vanishing volume [48, 49].”
- Homoclinic and heteroclinic bifurcations in vector fields (2010, unknown): “We refer to [6, 12, 15, 262] for further results, addressing primarily ergodic properties, Lorenz-like and other singular hyperbolic attractors.”