40,414 research outputs found
Recurrent flow analysis in spatiotemporally chaotic 2-dimensional Kolmogorov flow
Motivated by recent success in the dynamical systems approach to transitional
flow, we study the efficiency and effectiveness of extracting simple invariant
sets (recurrent flows) directly from chaotic/turbulent flows and the potential
of these sets for providing predictions of certain statistics of the flow.
Two-dimensional Kolmogorov flow (the 2D Navier-Stokes equations with a
sinusoidal body force) is studied both over a square [0, 2{\pi}]2 torus and a
rectangular torus extended in the forcing direction. In the former case, an
order of magnitude more recurrent flows are found than previously (Chandler &
Kerswell 2013) and shown to give improved predictions for the dissipation and
energy pdfs of the chaos via periodic orbit theory. Over the extended torus at
low forcing amplitudes, some extracted states mimick the statistics of the
spatially-localised chaos present surprisingly well recalling the striking
finding of Kawahara & Kida (2001) in low-Reynolds-number plane Couette flow. At
higher forcing amplitudes, however, success is limited highlighting the
increased dimensionality of the chaos and the need for larger data sets.
Algorithmic developments to improve the extraction procedure are discussed
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
Return-Map Cryptanalysis Revisited
As a powerful cryptanalysis tool, the method of return-map attacks can be
used to extract secret messages masked by chaos in secure communication
schemes. Recently, a simple defensive mechanism was presented to enhance the
security of chaotic parameter modulation schemes against return-map attacks.
Two techniques are combined in the proposed defensive mechanism: multistep
parameter modulation and alternative driving of two different transmitter
variables. This paper re-studies the security of this proposed defensive
mechanism against return-map attacks, and points out that the security was much
over-estimated in the original publication for both ciphertext-only attack and
known/chosen-plaintext attacks. It is found that a deterministic relationship
exists between the shape of the return map and the modulated parameter, and
that such a relationship can be used to dramatically enhance return-map attacks
thereby making them quite easy to break the defensive mechanism.Comment: 11 pages, 7 figure
Inverting Chaos: Extracting System Parameters from Experimental Data
Given a set of experimental or numerical chaotic data and a set of model differential equations with several parameters, is it possible to determine the numerical values for these parameters using a least-squares approach, and thereby to test the model against the data? We explore this question (a) with simulated data from model equations for the Rossler, Lorenz, and pendulum attractors, and (b) with experimental data produced by a physical chaotic pendulum. For the systems considered in this paper, the least-squares approach provides values of model parameters that agree well with values obtained in other ways, even in the presence of modest amounts of added noise. For experimental data, the “fitted” and experimental attractors are found to have the same correlation dimension and the same positive Lyapunov exponent
Extracting Lyapunov exponents from the echo dynamics of Bose-Einstein condensates on a lattice
We propose theoretically an experimentally realizable method to demonstrate
the Lyapunov instability and to extract the value of the largest Lyapunov
exponent for a chaotic many-particle interacting system. The proposal focuses
specifically on a lattice of coupled Bose-Einstein condensates in the classical
regime describable by the discrete Gross-Pitaevskii equation. We suggest to use
imperfect time-reversal of system's dynamics known as Loschmidt echo, which can
be realized experimentally by reversing the sign of the Hamiltonian of the
system. The routine involves tracking and then subtracting the noise of
virtually any observable quantity before and after the time-reversal. We
support the theoretical analysis by direct numerical simulations demonstrating
that the largest Lyapunov exponent can indeed be extracted from the Loschmidt
echo routine. We also discuss possible values of experimental parameters
required for implementing this proposal
Universal rank-order transform to extract signals from noisy data
We introduce an ordinate method for noisy data analysis, based solely on rank information and thus insensitive to outliers. The method is nonparametric and objective, and the required data processing is parsimonious. The main ingredients include a rank-order data matrix and its transform to a stable form, which provide linear trends in excellent agreement with least squares regression, despite the loss of magnitude information. A group symmetry orthogonal decomposition of the 2D rank-order transform for iid (white) noise is further ordered by principal component analysis. This two-step procedure provides a noise “etalon” used to characterize arbitrary stationary stochastic processes. The method readily distinguishes both the Ornstein-Uhlenbeck process and chaos generated by the logistic map from white noise. Ranking within randomness differs fundamentally from that in deterministic chaos and signals, thus forming the basis for signal detection. To further illustrate the breadth of applications, we apply this ordinate method to the canonical nonlinear parameter estimation problem of two-species radioactive decay, outperforming special-purpose least squares software. We demonstrate that the method excels when extracting trends in heavy-tailed noise and, unlike the Thiele-Sen estimator, is not limited to linear regression. A simple expression is given that yields a close approximation for signal extraction of an underlying, generally nonlinear signal
Breaking a chaos-based secure communication scheme designed by an improved modulation method
Recently Bu and Wang [Chaos, Solitons & Fractals 19 (2004) 919] proposed a
simple modulation method aiming to improve the security of chaos-based secure
communications against return-map-based attacks. Soon this modulation method
was independently cryptanalyzed by Chee et al. [Chaos, Solitons & Fractals 21
(2004) 1129], Wu et al. [Chaos, Solitons & Fractals 22 (2004) 367], and
\'{A}lvarez et al. [Chaos, Solitons & Fractals, accepted (2004),
arXiv:nlin.CD/0406065] via different attacks. As an enhancement to the Bu-Wang
method, an improving scheme was suggested by Wu et al. by removing the
relationship between the modulating function and the zero-points. The present
paper points out that the improved scheme proposed by Wu et al. is still
insecure against a new attack. Compared with the existing attacks, the proposed
attack is more powerful and can also break the original Bu-Wang scheme.
Furthermore, it is pointed out that the security of the modulation-based
schemes is not so satisfactory from a pure cryptographical point of view. The
synchronization performance of this class of modulation-based schemes is also
discussed.Comment: elsart.cls, 18 pages, 9 figure
- …