85 research outputs found
Linear response formula for piecewise expanding unimodal maps
The average R(t) of a smooth function with respect to the SRB measure of a
smooth one-parameter family f_t of piecewise expanding interval maps is not
always Lipschitz. We prove that if f_t is tangent to the topological class of
f_0, then R(t) is differentiable at zero, and the derivative coincides with the
resummation previously proposed by the first named author of the (a priori
divergent) series given by Ruelle's conjecture.Comment: We added Theorem 7.1 which shows that the horizontality condition is
necessary. The paper "Smooth deformations..." containing Thm 2.8 is now
available on the arxiv; see also Corrigendum arXiv:1205.5468 (to appear
Nonlinearity 2012
Dissipation time and decay of correlations
We consider the effect of noise on the dynamics generated by
volume-preserving maps on a d-dimensional torus. The quantity we use to measure
the irreversibility of the dynamics is the dissipation time. We focus on the
asymptotic behaviour of this time in the limit of small noise. We derive
universal lower and upper bounds for the dissipation time in terms of various
properties of the map and its associated propagators: spectral properties,
local expansivity, and global mixing properties. We show that the dissipation
is slow for a general class of non-weakly-mixing maps; on the opposite, it is
fast for a large class of exponentially mixing systems which include uniformly
expanding maps and Anosov diffeomorphisms.Comment: 26 Pages, LaTex. Submitted to Nonlinearit
Effects of the Bacterial Extract OM-85 on Phagocyte Functions and the Stress Response
The effects of the bacterial extract OM-85 on the respiratory burst,
intracellular calcium and the stress response have been investigated
in human peripheral blood monocytes from normal donors. Activation
of the respiratory burst during bacterial phagocytosis has been
previously associated with heat shock/stress proteins synthesis.
Whereas OM-85 stimulated superoxide production and increased
Ca2+ mobilization, it fared to induce synthesis of
classical HSPs. The lack of stress protein induction was observed
even in the presence of iron which potentiates both oxidative injury
and stress protein induction during bacterial phagocytosis. However
OM-85 induced a 75–78 kDa protein, which is likely to be a
glucose regulated protein (GRP78), and enhanced intracellular
expression of interleukin-lβ precursor
Convergence of invariant densities in the small-noise limit
This paper presents a systematic numerical study of the effects of noise on
the invariant probability densities of dynamical systems with varying degrees
of hyperbolicity. It is found that the rate of convergence of invariant
densities in the small-noise limit is frequently governed by power laws. In
addition, a simple heuristic is proposed and found to correctly predict the
power law exponent in exponentially mixing systems. In systems which are not
exponentially mixing, the heuristic provides only an upper bound on the power
law exponent. As this numerical study requires the computation of invariant
densities across more than 2 decades of noise amplitudes, it also provides an
opportunity to discuss and compare standard numerical methods for computing
invariant probability densities.Comment: 27 pages, 19 figures, revised with minor correction
Rare events, escape rates and quasistationarity: some exact formulae
We present a common framework to study decay and exchanges rates in a wide
class of dynamical systems. Several applications, ranging form the metric
theory of continuons fractions and the Shannon capacity of contrained systems
to the decay rate of metastable states, are given
A quantization procedure based on completely positive maps and Markov operators
We describe -limit sets of completely positive (CP) maps over
finite-dimensional spaces. In such sets and in its corresponding convex hulls,
CP maps present isometric behavior and the states contained in it commute with
each other. Motivated by these facts, we describe a quantization procedure
based on CP maps which are induced by Markov (transfer) operators. Classical
dynamics are described by an action over essentially bounded functions. A
non-expansive linear map, which depends on a choice of a probability measure,
is the centerpiece connecting phenomena over function and matrix spaces
Analyticity of the SRB measure of a lattice of coupled Anosov diffeomorphisms of the torus
We consider the "thermodynamic limit"of a d-dimensional lattice of hyperbolic
dynamical systems on the 2-torus, interacting via weak and nearest neighbor
coupling. We prove that the SRB measure is analytic in the strength of the
coupling. The proof is based on symbolic dynamics techniques that allow us to
map the SRB measure into a Gibbs measure for a spin system on a
(d+1)-dimensional lattice. This Gibbs measure can be studied by an extension
(decimation) of the usual "cluster expansion" techniques.Comment: 28 pages, 2 figure
Ruelle-Perron-Frobenius spectrum for Anosov maps
We extend a number of results from one dimensional dynamics based on spectral
properties of the Ruelle-Perron-Frobenius transfer operator to Anosov
diffeomorphisms on compact manifolds. This allows to develop a direct operator
approach to study ergodic properties of these maps. In particular, we show that
it is possible to define Banach spaces on which the transfer operator is
quasicompact. (Information on the existence of an SRB measure, its smoothness
properties and statistical properties readily follow from such a result.) In
dimension we show that the transfer operator associated to smooth random
perturbations of the map is close, in a proper sense, to the unperturbed
transfer operator. This allows to obtain easily very strong spectral stability
results, which in turn imply spectral stability results for smooth
deterministic perturbations as well. Finally, we are able to implement an Ulam
type finite rank approximation scheme thus reducing the study of the spectral
properties of the transfer operator to a finite dimensional problem.Comment: 58 pages, LaTe
Spectral properties of noisy classical and quantum propagators
We study classical and quantum maps on the torus phase space, in the presence
of noise. We focus on the spectral properties of the noisy evolution operator,
and prove that for any amount of noise, the quantum spectrum converges to the
classical one in the semiclassical limit. The small-noise behaviour of the
classical spectrum highly depends on the dynamics generated by the map. For a
chaotic dynamics, the outer spectrum consists in isolated eigenvalues
(``resonances'') inside the unit circle, leading to an exponential damping of
correlations. On the opposite, in the case of a regular map, part of the
spectrum accumulates along a one-dimensional ``string'' connecting the origin
with unity, yielding a diffusive behaviour. We finally study the
non-commutativity between the semiclassical and small-noise limits, and
illustrate this phenomenon by computing (analytically and numerically) the
classical and quantum spectra for some maps.Comment: 35 pages, 6 .eps figures, to be published in Nonlinearity. I added
some references and comment
Statistical and dynamical properties of covariant lyapunov vectors in a coupled atmosphere-ocean model—multiscale effects, geometric degeneracy, and error dynamics
We study a simplified coupled atmosphere-ocean model using the formalism of covariant Lyapunov vectors (CLVs), which link physically-based directions of perturbations to growth/decay rates. The model is obtained via a severe truncation of quasi-geostrophic equations for the two fluids, and includes a simple yet physically meaningful representation of their dynamical/thermodynamical coupling. The model has 36 degrees of freedom, and the parameters are chosen so that a chaotic behaviour is observed. There are two positive Lyapunov exponents (LEs), sixteen negative LEs, and eighteen near-zero LEs. The presence of many near-zero LEs results from the vast time-scale separation between the characteristic time scales of the two fluids, and leads to nontrivial error growth properties in the tangent space spanned by the corresponding CLVs, which are geometrically very degenerate. Such CLVs correspond to two different classes of ocean/atmosphere coupled modes. The tangent space spanned by the CLVs corresponding to the positive and negative LEs has, instead, a non-pathological behaviour, and one can construct robust large deviations laws for the finite time LEs, thus providing a universal model for assessing predictability on long to ultra-long scales along such directions. Interestingly, the tangent space of the unstable manifold has substantial projection on both atmospheric and oceanic components. The results show the difficulties in using hyperbolicity as a conceptual framework for multiscale chaotic dynamical systems, whereas the framework of partial hyperbolicity seems better suited, possibly indicating an alternative definition for the chaotic hypothesis. They also suggest the need for an accurate analysis of error dynamics on different time scales and domains and for a careful set-up of assimilation schemes when looking at coupled atmosphere-ocean models
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