23 research outputs found
Borel Cantelli Lemmas and Extreme Value Theory for Geometric Lorenz Models
We establish dynamical Borel-Cantelli lemmas for nested balls and rectangles
centered at generic points in the setting of geometric Lorenz maps. We also
establish extreme value statistics for observations maximized at generic points
for geometric Lorenz maps and the associated flow
Mixing properties and statistical limit theorems for singular hyperbolic flows without a smooth stable foliation
Over the last 10 years or so, advanced statistical properties, including
exponential decay of correlations, have been established for certain classes of
singular hyperbolic flows in three dimensions. The results apply in particular
to the classical Lorenz attractor. However, many of the proofs rely heavily on
the smoothness of the stable foliation for the flow.
In this paper, we show that many statistical properties hold for singular
hyperbolic flows with no smoothness assumption on the stable foliation. These
properties include existence of SRB measures, central limit theorems and
associated invariance principles, as well as results on mixing and rates of
mixing. The properties hold equally for singular hyperbolic flows in higher
dimensions provided the center-unstable subspaces are two-dimensional.Comment: Accepted version. To appear in Advances in Mat
On statistical attractors and the convergence of time averages
Copyright © 2011 Cambridge Philosophical SocietyThere are various notions of attractor in the literature, including measure (Milnor) attractors and statistical (Ilyashenko) attractors. In this paper we relate the notion of statistical attractor to that of the essential ω-limit set and prove some elementary results about these. In addition, we consider the convergence of time averages along trajectories. Ergodicity implies the convergence of time averages along almost all trajectories for all continuous observables. For non-ergodic systems, time averages may not exist even for almost all trajectories. However, averages of some observables may converge; we characterize conditions on observables that ensure convergence of time averages even in non-ergodic systems
From random Poincar\'e maps to stochastic mixed-mode-oscillation patterns
We quantify the effect of Gaussian white noise on fast--slow dynamical
systems with one fast and two slow variables, which display mixed-mode
oscillations owing to the presence of a folded-node singularity. The stochastic
system can be described by a continuous-space, discrete-time Markov chain,
recording the returns of sample paths to a Poincar\'e section. We provide
estimates on the kernel of this Markov chain, depending on the system
parameters and the noise intensity. These results yield predictions on the
observed random mixed-mode oscillation patterns. Our analysis shows that there
is an intricate interplay between the number of small-amplitude oscillations
and the global return mechanism. In combination with a local saturation
phenomenon near the folded node, this interplay can modify the number of
small-amplitude oscillations after a large-amplitude oscillation. Finally,
sufficient conditions are derived which determine when the noise increases the
number of small-amplitude oscillations and when it decreases this number.Comment: 56 pages, 14 figures; revised versio
Dominated splitting and zero volume for incompressible three-flows
We prove that there exists an open and dense subset of the incompressible
3-flows of class C^2 such that, if a flow in this set has a positive volume
regular invariant subset with dominated splitting for the linear Poincar\'e
flow, then it must be an Anosov flow. With this result we are able to extend
the dichotomies of Bochi-Ma\~n\'e and of Newhouse for flows with singularities.
That is we obtain for a residual subset of the C^1 incompressible flows on
3-manifolds that: (i) either all Lyapunov exponents are zero or the flow is
Anosov, and (ii) either the flow is Anosov or else the elliptic periodic points
are dense in the manifold.Comment: 23 pages, no figure