116 research outputs found
A phenomenological approach to normal form modeling: a case study in laser induced nematodynamics
An experimental setting for the polarimetric study of optically induced
dynamical behavior in nematic liquid crystal films has allowed to identify most
notably some behavior which was recognized as gluing bifurcations leading to
chaos. This analysis of the data used a comparison with a model for the
transition to chaos via gluing bifurcations in optically excited nematic liquid
crystals previously proposed by G. Demeter and L. Kramer. The model of these
last authors, proposed about twenty years before, does not have the central
symmetry which one would expect for minimal dimensional models for chaos in
nematics in view of the time series. What we show here is that the simplest
truncated normal forms for gluing, with the appropriate symmetry and minimal
dimension, do exhibit time signals that are embarrassingly similar to the ones
found using the above mentioned experimental settings. The gluing bifurcation
scenario itself is only visible in limited parameter ranges and substantial
aspect of the chaos that can be observed is due to other factors. First, out of
the immediate neighborhood of the homoclinic curve, nonlinearity can produce
expansion leading to chaos when combined with the recurrence induced by the
homoclinic behavior. Also, pairs of symmetric homoclinic orbits create extreme
sensitivity to noise, so that when the noiseless approach contains a rich
behavior, minute noise can transform the complex damping into sustained chaos.
Leonid Shil'nikov taught us that combining global considerations and local
spectral analysis near critical points is crucial to understand the
phenomenology associated to homoclinic bifurcations. Here this helps us
construct a phenomenological approach to modeling experiments in nonlinear
dissipative contexts.Comment: 25 pages, 9 figure
Period Doubling Renormalization for Area-Preserving Maps and Mild Computer Assistance in Contraction Mapping Principle
It has been observed that the famous Feigenbaum-Coullet-Tresser period
doubling universality has a counterpart for area-preserving maps of {\fR}^2.
A renormalization approach has been used in a "hard" computer-assisted proof of
existence of an area-preserving map with orbits of all binary periods in
Eckmann et al (1984). As it is the case with all non-trivial universality
problems in non-dissipative systems in dimensions more than one, no analytic
proof of this period doubling universality exists to date.
In this paper we attempt to reduce computer assistance in the argument, and
present a mild computer aided proof of the analyticity and compactness of the
renormalization operator in a neighborhood of a renormalization fixed point:
that is a proof that does not use generalizations of interval arithmetics to
functional spaces - but rather relies on interval arithmetics on real numbers
only to estimate otherwise explicit expressions. The proof relies on several
instance of the Contraction Mapping Principle, which is, again, verified via
mild computer assistance
Convex Dynamics and Applications
This paper proves a theorem about bounding orbits of a time dependent
dynamical system. The maps that are involved are examples in convex dynamics,
by which we mean the dynamics of piecewise isometries where the pieces are
convex. The theorem came to the attention of the authors in connection with the
problem of digital halftoning. \textit{Digital halftoning} is a family of
printing technologies for getting full color images from only a few different
colors deposited at dots all of the same size. The simplest version consist in
obtaining grey scale images from only black and white dots. A corollary of the
theorem is that for \textit{error diffusion}, one of the methods of digital
halftoning, averages of colors of the printed dots converge to averages of the
colors taken from the same dots of the actual images. Digital printing is a
special case of a much wider class of scheduling problems to which the theorem
applies. Convex dynamics has roots in classical areas of mathematics such as
symbolic dynamics, Diophantine approximation, and the theory of uniform
distributions.Comment: LaTex with 9 PostScript figure
Stably non-synchronizable maps of the plane
Pecora and Carroll presented a notion of synchronization where an
(n-1)-dimensional nonautonomous system is constructed from a given
-dimensional dynamical system by imposing the evolution of one coordinate.
They noticed that the resulting dynamics may be contracting even if the
original dynamics are not. It is easy to construct flows or maps such that no
coordinate has synchronizing properties, but this cannot be done in an open set
of linear maps or flows in , . In this paper we give examples of
real analytic homeomorphisms of such that the non-synchronizability is
stable in the sense that in a full neighborhood of the given map, no
homeomorphism is synchronizable
A numerical study of infinitely renormalizable area-preserving maps
It has been shown in (Gaidashev et al, 2010) and (Gaidashev et al, 2011) that
infinitely renormalizable area-preserving maps admit invariant Cantor sets with
a maximal Lyapunov exponent equal to zero. Furthermore, the dynamics on these
Cantor sets for any two infinitely renormalizable maps is conjugated by a
transformation that extends to a differentiable function whose derivative is
Holder continuous of exponent alpha>0.
In this paper we investigate numerically the specific value of alpha. We also
present numerical evidence that the normalized derivative cocycle with the base
dynamics in the Cantor set is ergodic. Finally, we compute renormalization
eigenvalues to a high accuracy to support a conjecture that the renormalization
spectrum is real
No extension of quantum theory can have improved predictive power
According to quantum theory, measurements generate random outcomes, in stark
contrast with classical mechanics. This raises the question of whether there
could exist an extension of the theory which removes this indeterminism, as
suspected by Einstein, Podolsky and Rosen (EPR). Although this has been shown
to be impossible, existing results do not imply that the current theory is
maximally informative. Here we ask the more general question of whether any
improved predictions can be achieved by any extension of quantum theory. Under
the assumption that measurements can be chosen freely, we answer this question
in the negative: no extension of quantum theory can give more information about
the outcomes of future measurements than quantum theory itself. Our result has
significance for the foundations of quantum mechanics, as well as applications
to tasks that exploit the inherent randomness in quantum theory, such as
quantum cryptography.Comment: 6 pages plus 7 of supplementary material, 3 figures. Title changed.
Added discussion on Bell's notion of locality. FAQ answered at
http://perimeterinstitute.ca/personal/rcolbeck/FAQ.htm
The boundary of chaos for interval mappings
A goal in the study of dynamics on the interval is to understand the transition to positive topological entropy. There is a conjecture from the 1980s that the only route to positive topological entropy is through a cascade of period doubling bifurcations. We prove this conjecture in natural families of smooth interval maps, and use it to study the structure of the boundary of mappings with positive entropy. In particular, we show that in families of mappings with a fixed number of critical points the boundary is locally connected, and for analytic mappings that it is a cellular set
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