142 research outputs found
Extreme Value laws for dynamical systems under observational noise
In this paper we prove the existence of Extreme Value Laws for dynamical
systems perturbed by instrument-like-error, also called observational noise. An
orbit perturbed with observational noise mimics the behavior of an
instrumentally recorded time series. Instrument characteristics - defined as
precision and accuracy - act both by truncating and randomly displacing the
real value of a measured observable. Here we analyze both these effects from a
theoretical and numerical point of view. First we show that classical extreme
value laws can be found for orbits of dynamical systems perturbed with
observational noise. Then we present numerical experiments to support the
theoretical findings and give an indication of the order of magnitude of the
instrumental perturbations which cause relevant deviations from the extreme
value laws observed in deterministic dynamical systems. Finally, we show that
the observational noise preserves the structure of the deterministic attractor.
This goes against the common assumption that random transformations cause the
orbits asymptotically fill the ambient space with a loss of information about
any fractal structures present on the attractor
Escape Rates Formulae and Metastability for Randomly perturbed maps
We provide escape rates formulae for piecewise expanding interval maps with
`random holes'. Then we obtain rigorous approximations of invariant densities
of randomly perturbed metabstable interval maps. We show that our escape rates
formulae can be used to approximate limits of invariant densities of randomly
perturbed metastable systems.Comment: Appeared in Nonlinearity, May 201
Metastability of Certain Intermittent Maps
We study an intermittent map which has exactly two ergodic invariant
densities. The densities are supported on two subintervals with a common
boundary point. Due to certain perturbations, leakage of mass through subsets,
called holes, of the initially invariant subintervals occurs and forces the
subsystems to merge into one system that has exactly one invariant density. We
prove that the invariant density of the perturbed system converges in the
-norm to a particular convex combination of the invariant densities of the
intermittent map. In particular, we show that the ratio of the weights in the
combination equals to the limit of the ratio of the measures of the holes.Comment: 19 pages, 2 figure
Annealed and quenched limit theorems for random expanding dynamical systems
In this paper, we investigate annealed and quenched limit theorems for random
expanding dynamical systems. Making use of functional analytic techniques and
more probabilistic arguments with martingales, we prove annealed versions of a
central limit theorem, a large deviation principle, a local limit theorem, and
an almost sure invariance principle. We also discuss the quenched central limit
theorem, dynamical Borel-Cantelli lemmas, Erd\"os-R\'enyi laws and
concentration inequalities.Comment: Appeared online in Probability Theory and Related Field
Mixing properties in the advection of passive tracers via recurrences and extreme value theory
In this paper we characterize the mixing properties in the advection of
passive tracers by exploiting the extreme value theory for dynamical systems.
With respect to classical techniques directly related to the Poincar\'e
recurrences analysis, our method provides reliable estimations of the
characteristic mixing times and distinguishes between barriers and unstable
fixed points. The method is based on a check of convergence for extreme value
laws on finite datasets. We define the mixing times in terms of the shortest
time intervals such that extremes converge to the asymptotic (known) parameters
of the Generalized Extreme Value distribution. Our technique is suitable for
applications in the analysis of other systems where mixing time scales need to
be determined and limited datasets are available.Comment: arXiv admin note: text overlap with arXiv:1107.597
Diffusion on the torus for Hamiltonian maps
For a mapping of the torusT2 we propose a definition of the diffusion coefficientD suggested by the solution of the diffusion equation ofT2. The definition ofD, based on the limit of moments of the invariant measure, depends on the set Ω where an initial uniform distribution is assigned. For the algebraic automorphism of the torus the limit is proved to exist and to have the same value for almost all initial sets Ω in the subfamily of parallelograms. Numerical results show that it has the same value for arbitrary polygons Ω and for arbitrary moments
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