15,983 research outputs found
An elementary approach to rigorous approximation of invariant measures
We describe a framework in which is possible to develop and implement
algorithms for the approximation of invariant measures of dynamical systems
with a given bound on the error of the approximation.
Our approach is based on a general statement on the approximation of fixed
points for operators between normed vector spaces, allowing an explicit
estimation of the error.
We show the flexibility of our approach by applying it to piecewise expanding
maps and to maps with indifferent fixed points. We show how the required
estimations can be implemented to compute invariant densities up to a given
error in the or distance. We also show how to use this to
compute an estimation with certified error for the entropy of those systems.
We show how several related computational and numerical issues can be solved
to obtain working implementations, and experimental results on some one
dimensional maps.Comment: 27 pages, 10 figures. Main changes: added a new section in which we
apply our method to Manneville-Pomeau map
An elementary way to rigorously estimate convergence to equilibrium and escape rates
We show an elementary method to have (finite time and asymptotic) computer
assisted explicit upper bounds on convergence to equilibrium (decay of
correlations) and escape rate for systems satisfying a Lasota Yorke inequality.
The bounds are deduced by the ones of suitable approximations of the system's
transfer operator. We also present some rigorous experiment showing the
approach and some concrete result.Comment: 14 pages, 6 figure
A Rigorous Computational Approach to Linear Response
We present a general setting in which the formula describing the linear
response of the physical measure of a perturbed system can be obtained. In this
general setting we obtain an algorithm to rigorously compute the linear
response. We apply our results to expanding circle maps. In particular, we
present examples where we compute, up to a pre-specified error in the
-norm, the response of expanding circle maps under stochastic and
deterministic perturbations. Moreover, we present an example where we compute,
up to a pre-specified error in the -norm, the response of the intermittent
family at the boundary; i.e., when the unperturbed system is the doubling map.Comment: Revised version following reports. A new example which contains the
computation of the linear response at the boundary of the intermittent family
has been adde
Rigorous approximation of diffusion coefficients for expanding maps
We use Ulam's method to provide rigorous approximation of diffusion
coefficients for uniformly expanding maps. An algorithm is provided and its
implementation is illustrated using Lanford's map.Comment: In this version Lanford's map has been used to illustrate the
computer implementation of the algorithm. To appear in Journal of Statistical
Physic
Statistical properties of dynamics. Introduction to the functional analytic approach
These are lecture notes for a simple minicourse approaching the satistical
properties of a dynamical system by the study of the associated transfer
operator (considered on a suitable function space). The following questions
will be addressed:
* existence of a regular invariant measure;
* Lasota Yorke inequalities and spectral gap;
* decay of correlations and some limit theorem;
* stability under perturbations of the system;
* linear response;
* hyperbolic systems.
The point of view taken is to present the general construction and ideas
needed to obtain these results in the simplest way. For this, some theorem is
proved in a form which is weaker than usually known, but with an elementary and
simple proof.
These notes are intended for the Hokkaido-Pisa University summer course 2017.Comment: I decided to make these lecture notes public because it will be cited
in some research paper. I hope these will be useful for some reader. In this
new version several new topics are added, with some original approac
Projection operator formalism and entropy
The entropy definition is deduced by means of (re)deriving the generalized
non-linear Langevin equation using Zwanzig projector operator formalism. It is
shown to be necessarily related to an invariant measure which, in classical
mechanics, can always be taken to be the Liouville measure. It is not true that
one is free to choose a ``relevant'' probability density independently as is
done in other flavors of projection operator formalism. This observation
induces an entropy expression which is valid also outside the thermodynamic
limit and in far from equilibrium situations. The Zwanzig projection operator
formalism therefore gives a deductive derivation of non-equilibrium, and
equilibrium, thermodynamics. The entropy definition found is closely related to
the (generalized) microcanonical Boltzmann-Planck definition but with some
subtle differences. No ``shell thickness'' arguments are needed, nor desirable,
for a rigorous definition. The entropy expression depends on the choice of
macroscopic variables and does not exactly transform as a scalar quantity. The
relation with expressions used in the GENERIC formalism are discussed
Spike trains statistics in Integrate and Fire Models: exact results
We briefly review and highlight the consequences of rigorous and exact
results obtained in \cite{cessac:10}, characterizing the statistics of spike
trains in a network of leaky Integrate-and-Fire neurons, where time is discrete
and where neurons are subject to noise, without restriction on the synaptic
weights connectivity. The main result is that spike trains statistics are
characterized by a Gibbs distribution, whose potential is explicitly
computable. This establishes, on one hand, a rigorous ground for the current
investigations attempting to characterize real spike trains data with Gibbs
distributions, such as the Ising-like distribution, using the maximal entropy
principle. However, it transpires from the present analysis that the Ising
model might be a rather weak approximation. Indeed, the Gibbs potential (the
formal "Hamiltonian") is the log of the so-called "conditional intensity" (the
probability that a neuron fires given the past of the whole network). But, in
the present example, this probability has an infinite memory, and the
corresponding process is non-Markovian (resp. the Gibbs potential has infinite
range). Moreover, causality implies that the conditional intensity does not
depend on the state of the neurons at the \textit{same time}, ruling out the
Ising model as a candidate for an exact characterization of spike trains
statistics. However, Markovian approximations can be proposed whose degree of
approximation can be rigorously controlled. In this setting, Ising model
appears as the "next step" after the Bernoulli model (independent neurons)
since it introduces spatial pairwise correlations, but not time correlations.
The range of validity of this approximation is discussed together with possible
approaches allowing to introduce time correlations, with algorithmic
extensions.Comment: 6 pages, submitted to conference NeuroComp2010
http://2010.neurocomp.fr/; Bruno Cessac
http://www-sop.inria.fr/neuromathcomp
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