328 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
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
Global vs local energy dissipation: the energy cycle of the turbulent von K\'arm\'an flow
In this paper, we investigate the relations between global and local energy
transfers in a turbulent von K\'arm\'an flow. The goal is to understand how and
where energy is dissipated in such a flow and to reconstruct the energy cycle
in an experimental device where local as well as global quantities can be
measured. We use PIV measurements and we model the Reynolds stress tensor to
take subgrid scales into account. This procedure involves a free parameter that
is calibrated using angular momentum balance. We then estimate the local and
global mean injected and dissipated power for several types of impellers, for
various Reynolds numbers and for various flow topologies. These PIV estimates
are then compared with direct injected power estimates provided by torque
measurements at the impellers. The agreement between PIV estimates and direct
measurements depends on the flow topology. In symmetric situations, we are able
to capture up to 90% of the actual global energy dissipation rate. However, our
results become increasingly inaccurate as the shear layer responsible for most
of the dissipation approaches one of the impellers, and cannot be resolved by
our PIV set-up. Finally, we show that a very good agreement between PIV
estimates and direct measurements is obtained using a new method based on the
work of Duchon and Robert which generalizes the K\'arm\'an-Howarth equation to
nonisotropic, nonhomogeneous flows. This method provides parameter-free
estimates of the energy dissipation rate as long as the smallest resolved scale
lies in the inertial range. These results are used to evidence a well-defined
stationary energy cycle within the flow in which most of the energy is injected
at the top and bottom impellers, and dissipated within the shear layer. The
influence of the mean flow geometry and the Reynolds number on this energy
cycle is studied for a wide range of parameters
Statistical early-warning indicators based on Auto-Regressive Moving-Average processes
We address the problem of defining early warning indicators of critical
transition. To this purpose, we fit the relevant time series through a class of
linear models, known as Auto-Regressive Moving-Average (ARMA(p,q)) models. We
define two indicators representing the total order and the total persistence of
the process, linked, respectively, to the shape and to the characteristic decay
time of the autocorrelation function of the process. We successfully test the
method to detect transitions in a Langevin model and a 2D Ising model with
nearest-neighbour interaction. We then apply the method to complex systems,
namely for dynamo thresholds and financial crisis detection.Comment: 5 pages, 4 figure
Extreme Value distribution for singular measures
In this paper we perform an analytical and numerical study of Extreme Value
distributions in discrete dynamical systems that have a singular measure. Using
the block maxima approach described in Faranda et al. [2011] we show that,
numerically, the Extreme Value distribution for these maps can be associated to
the Generalised Extreme Value family where the parameters scale with the
information dimension. The numerical analysis are performed on a few low
dimensional maps. For the middle third Cantor set and the Sierpinskij triangle
obtained using Iterated Function Systems, experimental parameters show a very
good agreement with the theoretical values. For strange attractors like Lozi
and H\`enon maps a slower convergence to the Generalised Extreme Value
distribution is observed. Even in presence of large statistics the observed
convergence is slower if compared with the maps which have an absolute
continuous invariant measure. Nevertheless and within the uncertainty computed
range, the results are in good agreement with the theoretical estimates
Towards a General Theory of Extremes for Observables of Chaotic Dynamical Systems
In this paper we provide a connection between the geometrical properties of a
chaotic dynamical system and the distribution of extreme values. We show that
the extremes of so-called physical observables are distributed according to the
classical generalised Pareto distribution and derive explicit expressions for
the scaling and the shape parameter. In particular, we derive that the shape
parameter does not depend on the chosen observables, but only on the partial
dimensions of the invariant measure on the stable, unstable, and neutral
manifolds. The shape parameter is negative and is close to zero when
high-dimensional systems are considered. This result agrees with what was
derived recently using the generalized extreme value approach. Combining the
results obtained using such physical observables and the properties of the
extremes of distance observables, it is possible to derive estimates of the
partial dimensions of the attractor along the stable and the unstable
directions of the flow. Moreover, by writing the shape parameter in terms of
moments of the extremes of the considered observable and by using linear
response theory, we relate the sensitivity to perturbations of the shape
parameter to the sensitivity of the moments, of the partial dimensions, and of
the Kaplan-Yorke dimension of the attractor. Preliminary numerical
investigations provide encouraging results on the applicability of the theory
presented here. The results presented here do not apply for all combinations of
Axiom A systems and observables, but the breakdown seems to be related to very
special geometrical configurations.Comment: 16 pages, 3 Figure
Statistical optimization for passive scalar transport: maximum entropy production vs maximum Kolmogorov-Sinay entropy
We derive rigorous results on the link between the principle of maximum
entropy production and the principle of maximum Kolmogorov-Sinai entropy using
a Markov model of the passive scalar diffusion called the Zero Range Process.
We show analytically that both the entropy production and the Kolmogorov-Sinai
entropy seen as functions of f admit a unique maximum denoted fmaxEP and
fmaxKS. The behavior of these two maxima is explored as a function of the
system disequilibrium and the system resolution N. The main result of this
article is that fmaxEP and fmaxKS have the same Taylor expansion at _rst order
in the deviation of equilibrium. We find that fmaxEP hardly depends on N
whereas fmaxKS depends strongly on N. In particular, for a fixed difference of
potential between the reservoirs, fmaxEP (N) tends towards a non-zero value,
while fmaxKS (N) tends to 0 when N goes to infinity. For values of N typical of
that adopted by Paltridge and climatologists we show that fmaxEP and fmaxKS
coincide even far from equilibrium. Finally, we show that one can find an
optimal resolution N_ such that fmaxEP and fmaxKS coincide, at least up to a
second order parameter proportional to the non-equilibrium uxes imposed to the
boundaries.Comment: Nonlinear Processes in Geophysics (2015
Sampling local properties of attractors via Extreme Value Theory
We provide formulas to compute the coefficients entering the affine scaling
needed to get a non-degenerate function for the asymptotic distribution of the
maxima of some kind of observable computed along the orbit of a randomly
perturbed dynamical system. This will give information on the local geometrical
properties of the stationary measure. We will consider systems perturbed with
additive noise and with observational noise. Moreover we will apply our
techniques to chaotic systems and to contractive systems, showing that both
share the same qualitative behavior when perturbed
Probing turbulence intermittency via Auto-Regressive Moving-Average models
We suggest a new approach to probing intermittency corrections to the
Kolmogorov law in turbulent flows based on the Auto-Regressive Moving-Average
modeling of turbulent time series. We introduce a new index that
measures the distance from a Kolmogorov-Obukhov model in the Auto-Regressive
Moving-Average models space. Applying our analysis to Particle Image
Velocimetry and Laser Doppler Velocimetry measurements in a von K\'arm\'an
swirling flow, we show that is proportional to the traditional
intermittency correction computed from the structure function. Therefore it
provides the same information, using much shorter time series. We conclude that
is a suitable index to reconstruct the spatial intermittency of the
dissipation in both numerical and experimental turbulent fields.Comment: 5 page
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