20,555 research outputs found
Random Recurrent Neural Networks Dynamics
This paper is a review dealing with the study of large size random recurrent
neural networks. The connection weights are selected according to a probability
law and it is possible to predict the network dynamics at a macroscopic scale
using an averaging principle. After a first introductory section, the section 1
reviews the various models from the points of view of the single neuron
dynamics and of the global network dynamics. A summary of notations is
presented, which is quite helpful for the sequel. In section 2, mean-field
dynamics is developed.
The probability distribution characterizing global dynamics is computed. In
section 3, some applications of mean-field theory to the prediction of chaotic
regime for Analog Formal Random Recurrent Neural Networks (AFRRNN) are
displayed. The case of AFRRNN with an homogeneous population of neurons is
studied in section 4. Then, a two-population model is studied in section 5. The
occurrence of a cyclo-stationary chaos is displayed using the results of
\cite{Dauce01}. In section 6, an insight of the application of mean-field
theory to IF networks is given using the results of \cite{BrunelHakim99}.Comment: Review paper, 36 pages, 5 figure
Periodic orbit spectrum in terms of Ruelle--Pollicott resonances
Fully chaotic Hamiltonian systems possess an infinite number of classical
solutions which are periodic, e.g. a trajectory ``p'' returns to its initial
conditions after some fixed time tau_p. Our aim is to investigate the spectrum
tau_1, tau_2, ... of periods of the periodic orbits. An explicit formula for
the density rho(tau) = sum_p delta (tau - tau_p) is derived in terms of the
eigenvalues of the classical evolution operator. The density is naturally
decomposed into a smooth part plus an interferent sum over oscillatory terms.
The frequencies of the oscillatory terms are given by the imaginary part of the
complex eigenvalues (Ruelle--Pollicott resonances). For large periods,
corrections to the well--known exponential growth of the smooth part of the
density are obtained. An alternative formula for rho(tau) in terms of the zeros
and poles of the Ruelle zeta function is also discussed. The results are
illustrated with the geodesic motion in billiards of constant negative
curvature. Connections with the statistical properties of the corresponding
quantum eigenvalues, random matrix theory and discrete maps are also
considered. In particular, a random matrix conjecture is proposed for the
eigenvalues of the classical evolution operator of chaotic billiards
A Survey on the Classical Limit of Quantum Dynamical Entropies
We analyze the behavior of quantum dynamical entropies production from
sequences of quantum approximants approaching their (chaotic) classical limit.
The model of the quantized hyperbolic automorphisms of the 2-torus is examined
in detail and a semi-classical analysis is performed on it using coherent
states, fulfilling an appropriate dynamical localization property.
Correspondence between quantum dynamical entropies and the Kolmogorov-Sinai
invariant is found only over time scales that are logarithmic in the
quantization parameter.Comment: LaTeX, 21 pages, Presented at the 3rd Workshop on Quantum Chaos and
Localization Phenomena, Warsaw, Poland, May 25-27, 200
A Computational Model for Quantum Measurement
Is the dynamical evolution of physical systems objectively a manifestation of
information processing by the universe? We find that an affirmative answer has
important consequences for the measurement problem. In particular, we calculate
the amount of quantum information processing involved in the evolution of
physical systems, assuming a finite degree of fine-graining of Hilbert space.
This assumption is shown to imply that there is a finite capacity to sustain
the immense entanglement that measurement entails. When this capacity is
overwhelmed, the system's unitary evolution becomes computationally unstable
and the system suffers an information transition (`collapse'). Classical
behaviour arises from the rapid cycles of unitary evolution and information
transitions.
Thus, the fine-graining of Hilbert space determines the location of the
`Heisenberg cut', the mesoscopic threshold separating the microscopic, quantum
system from the macroscopic, classical environment. The model can be viewed as
a probablistic complement to decoherence, that completes the measurement
process by turning decohered improper mixtures of states into proper mixtures.
It is shown to provide a natural resolution to the measurement problem and the
basis problem.Comment: 24 pages; REVTeX4; published versio
The enhanced Sanov theorem and propagation of chaos
We establish a Sanov type large deviation principle for an ensemble of
interacting Brownian rough paths. As application a large deviations for the
(-layer, enhanced) empirical measure of weakly interacting diffusions is
obtained. This in turn implies a propagation of chaos result in rough path
spaces and allows for a robust subsequent analysis of the particle system and
its McKean-Vlasov type limit, as shown in two corollaries.Comment: 42 page
The exponential map is chaotic: An invitation to transcendental dynamics
We present an elementary and conceptual proof that the complex exponential
map is "chaotic" when considered as a dynamical system on the complex plane.
(This result was conjectured by Fatou in 1926 and first proved by Misiurewicz
55 years later.) The only background required is a first undergraduate course
in complex analysis.Comment: 22 pages, 4 figures. (Provisionally) accepted for publication the
American Mathematical Monthly. V2: Final pre-publication version. The article
has been revised, corrected and shortened by 14 pages; see Version 1 for a
more detailed discussion of further properties of the exponential map and
wider transcendental dynamic
On the Lebesgue measure of Li-Yorke pairs for interval maps
We investigate the prevalence of Li-Yorke pairs for and
multimodal maps with non-flat critical points. We show that every
measurable scrambled set has zero Lebesgue measure and that all strongly
wandering sets have zero Lebesgue measure, as does the set of pairs of
asymptotic (but not asymptotically periodic) points.
If is topologically mixing and has no Cantor attractor, then typical
(w.r.t. two-dimensional Lebesgue measure) pairs are Li-Yorke; if additionally
admits an absolutely continuous invariant probability measure (acip), then
typical pairs have a dense orbit for . These results make use of
so-called nice neighborhoods of the critical set of general multimodal maps,
and hence uniformly expanding Markov induced maps, the existence of either is
proved in this paper as well.
For the setting where has a Cantor attractor, we present a trichotomy
explaining when the set of Li-Yorke pairs and distal pairs have positive
two-dimensional Lebesgue measure.Comment: 41 pages, 3 figure
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