Existence and smoothness of the stable foliation for sectional hyperbolic attractors

We prove the existence of a contracting invariant topological foliation in a full neighborhood for partially hyperbolic attractors. Under certain bunching conditions it can then be shown that this stable foliation is smooth. Specialising to sectional hyperbolic attractors, we give a verifiable condition for bunching. In particular, we show that the stable foliation for the classical Lorenz equation (and nearby vector fields) is better than $C^1$ which is crucial for recent results on exponential decay of correlations. In fact the foliation is at least $C^{1.278}$.Comment: Corrected estimate for smoothness of stable foliation. Clarification of which results hold for general partially hyperbolic attractors. Some minor typos fixed. Accepted for publication in Bull. London Math. So

Bifurcation from relative periodic solutions

Published versio

Averaging and rates of averaging for uniform families of deterministic fast-slow skew product systems

We consider families of fast-slow skew product maps of the form \begin{align*} x_{n+1} = x_n+\epsilon a(x_n,y_n,\epsilon), \quad y_{n+1} = T_\epsilon y_n, \end{align*} where $T_\epsilon$ is a family of nonuniformly expanding maps, and prove averaging and rates of averaging for the slow variables $x$ as $\epsilon\to0$. Similar results are obtained also for continuous time systems \begin{align*} \dot x = \epsilon a(x,y,\epsilon), \quad \dot y = g_\epsilon(y). \end{align*} Our results include cases where the family of fast dynamical systems consists of intermittent maps, unimodal maps (along the Collet-Eckmann parameters) and Viana maps.Comment: Shortened version. First order averaging moved into a remark. Explicit coupling argument moved into a separate not

On the Validity of the 0-1 Test for Chaos

In this paper, we present a theoretical justification of the 0-1 test for chaos. In particular, we show that with probability one, the test yields 0 for periodic and quasiperiodic dynamics, and 1 for sufficiently chaotic dynamics

Statistical Properties and Decay of Correlations for Interval Maps with Critical Points and Singularities

We consider a class of piecewise smooth one-dimensional maps with critical points and singularities (possibly with infinite derivative). Under mild summability conditions on the growth of the derivative on critical orbits, we prove the central limit theorem and a vector-valued almost sure invariance principle. We also obtain results on decay of correlations.Comment: 18 pages, minor revisions, to appear in Communications in Mathematical Physic

The effect of symmetry breaking on the dynamics near a structurally stable heteroclinic cycle between equilibria and a periodic orbit

The effect of small forced symmetry breaking on the dynamics near a structurally stable heteroclinic cycle connecting two equilibria and a periodic orbit is investigated. This type of system is known to exhibit complicated, possibly chaotic dynamics including irregular switching of sign of various phase space variables, but details of the mechanisms underlying the complicated dynamics have not previously been investigated. We identify global bifurcations that induce the onset of chaotic dynamics and switching near a heteroclinic cycle of this type, and by construction and analysis of approximate return maps, locate the global bifurcations in parameter space. We find there is a threshold in the size of certain symmetry-breaking terms below which there can be no persistent switching. Our results are illustrated by a numerical example

Future Cascadia megathrust rupture delineated by episodic tremor and slip

A suite of 15 episodic tremor and slip events imaged between 1997 and 2008 along the northern Cascadia subduction zone suggests future coseismic rupture will extend to 25 km depth, or approximately 60 km inland of the Pacific coast, rather than stopping offshore at 15 km depth. An ETS-derived coupling profile accurately predicts GPS measured interseismic deformation of the overlying North American plate, as measured by approximately 50 continuous GPS stations across western Washington State. When extrapolated over the 550-year average recurrence interval of Cascadia megathrust events, the coupling model also replicates the pattern and amplitude of coseismic coastal subsidence inferred from previous megathrust earthquakes here. For only the Washington State segment of the Cascadia margin, this translates into an Mw = 8.9 earthquake, with significant moment release close to its metropolitan centers

Episodic Tremor and Slip in the Pacific Northwest

Every 14 months the Pacific Northwest experiences slow slip on a fault that is the equivalent of about a magnitude 6.5 earthquake. While a typical earthquake of this magnitude happens in less than 10 seconds, the duration of these slip events is two to several weeks. The most recent event occurred from January 14 through February 1, 2007