Three-Dimensional Flows

Decay of correlations and laws of rare events for transitive random maps

“…Finally, Tucker [38,39] proved the existence and robustness of the Lorenz attractor and, as a consequence of the method of his proof, showed that these models do describe the behaviour of (3.1). For more information on the history of the subject and the construction of the geometric models, we refer the reader to Araujo, Pacifico and Viana [4,40] and references therein.…”

Geometric Lorenz Attractor and Orbital Shadowing Property

“…Now one can makes the geometric Lorenz attractor by using suspension flow as in Figure 2 (for more details see [1,2]). Komuro [3] proved geometric Lorenz attractor does not satisfy the shadowing property except the case of f (0) = 1 and f (1) = 1.…”

“…Numerical observation indicates that all solutions of the Lorenz system pass transversely through a square which will be denoted by Σ, and so a two-dimensional invertible Poincaré map F can be defined on Σ as in Figure 1 (for more details, see [1]). …”

Dominated splitting and zero volume for incompressible three flows

“…For the angle bound we argue by contradiction as in [28]: assume there [19] and [3,Appendix]) an arbitrarily small C 1 perturbation Z n of Y n , for all big enough n ≥ 1, sending the stable direction close to the unstable direction along the periodic orbit, such that the orbit of O n becomes a sink or a source for Z n . This contradicts the first part of the statement of the lemma.…”

“…[27,3]. This means that the continuation σ Z of the singularity has an associated homoclinic orbit γ such that W cu (σ Z ) intersects W s (σ Z ) along γ but not transversely, and γ ∩W ss (σ Z ) = / 0.…”