4,477 research outputs found
A novel pseudo-random number generator based on discrete chaotic iterations
Security of information transmitted through the Internet, against passive or
active attacks is an international concern. The use of a chaos-based
pseudo-random bit sequence to make it unrecognizable by an intruder, is a field
of research in full expansion. This mask of useful information by modulation or
encryption is a fundamental part of the TLS Internet exchange protocol. In this
paper, a new method using discrete chaotic iterations to generate pseudo-random
numbers is presented. This pseudo-random number generator has successfully
passed the NIST statistical test suite (NIST SP800-22). Security analysis shows
its good characteristics. The application for secure image transmission through
the Internet is proposed at the end of the paper.Comment: The First International Conference on Evolving Internet:Internet 2009
pp.71--76 http://dx.doi.org/10.1109/INTERNET.2009.1
Dynamics of Coupled Maps with a Conservation Law
A particularly simple model belonging to a wide class of coupled maps which
obey a local conservation law is studied. The phase structure of the system and
the types of the phase transitions are determined. It is argued that the
structure of the phase diagram is robust with respect to mild violations of the
conservation law. Critical exponents possibly determining a new universality
class are calculated for a set of independent order parameters. Numerical
evidence is produced suggesting that the singularity in the density of Lyapunov
exponents at is a reflection of the singularity in the density of
Fourier modes (a ``Van Hove'' singularity) and disappears if the conservation
law is broken. Applicability of the Lyapunov dimension to the description of
spatiotemporal chaos in a system with a conservation law is discussed.Comment: To be published in CHAOS #7 (31 page, 16 figures
Theoretical Design and FPGA-Based Implementation of Higher-Dimensional Digital Chaotic Systems
Traditionally, chaotic systems are built on the domain of infinite precision
in mathematics. However, the quantization is inevitable for any digital
devices, which causes dynamical degradation. To cope with this problem, many
methods were proposed, such as perturbing chaotic states and cascading multiple
chaotic systems. This paper aims at developing a novel methodology to design
the higher-dimensional digital chaotic systems (HDDCS) in the domain of finite
precision. The proposed system is based on the chaos generation strategy
controlled by random sequences. It is proven to satisfy the Devaney's
definition of chaos. Also, we calculate the Lyapunov exponents for HDDCS. The
application of HDDCS in image encryption is demonstrated via FPGA platform. As
each operation of HDDCS is executed in the same fixed precision, no
quantization loss occurs. Therefore, it provides a perfect solution to the
dynamical degradation of digital chaos.Comment: 12 page
Time Quasilattices in Dissipative Dynamical Systems
We establish the existence of `time quasilattices' as stable trajectories in
dissipative dynamical systems. These tilings of the time axis, with two unit
cells of different durations, can be generated as cuts through a periodic
lattice spanned by two orthogonal directions of time. We show that there are
precisely two admissible time quasilattices, which we term the infinite Pell
and Clapeyron words, reached by a generalization of the period-doubling
cascade. Finite Pell and Clapeyron words of increasing length provide
systematic periodic approximations to time quasilattices which can be verified
experimentally. The results apply to all systems featuring the universal
sequence of periodic windows. We provide examples of discrete-time maps, and
periodically-driven continuous-time dynamical systems. We identify quantum
many-body systems in which time quasilattices develop rigidity via the
interaction of many degrees of freedom, thus constituting dissipative discrete
`time quasicrystals'.Comment: 38 pages, 14 figures. This version incorporates "Pell and Clapeyron
Words as Stable Trajectories in Dynamical Systems", arXiv:1707.09333.
Submission to SciPos
Resonances of the Frobenius-Perron Operator for a Hamiltonian Map with a Mixed Phase Space
Resonances of the (Frobenius-Perron) evolution operator P for phase-space
densities have recently attracted considerable attention, in the context of
interrelations between classical and quantum dynamics. We determine these
resonances as well as eigenvalues of P for Hamiltonian systems with a mixed
phase space, by truncating P to finite size in a Hilbert space of phase-space
functions and then diagonalizing. The corresponding eigenfunctions are
localized on unstable manifolds of hyperbolic periodic orbits for resonances
and on islands of regular motion for eigenvalues. Using information drawn from
the eigenfunctions we reproduce the resonances found by diagonalization through
a variant of the cycle expansion of periodic-orbit theory and as rates of
correlation decay.Comment: 18 pages, 7 figure
Open circle maps: Small hole asymptotics
We consider escape from chaotic maps through a subset of phase space, the
hole. Escape rates are known to be locally constant functions of the hole
position and size. In spite of this, for the doubling map we can extend the
current best result for small holes, a linear dependence on hole size h, to
include a smooth h^2 ln h term and explicit fractal terms to h^2 and higher
orders, confirmed by numerical simulations. For more general hole locations the
asymptotic form depends on a dynamical Diophantine condition using periodic
orbits ordered by stability.Comment: This version has a new section investigating different hole
locations. Now 9 pages, 3 figure
Predictability: a way to characterize Complexity
Different aspects of the predictability problem in dynamical systems are
reviewed. The deep relation among Lyapunov exponents, Kolmogorov-Sinai entropy,
Shannon entropy and algorithmic complexity is discussed. In particular, we
emphasize how a characterization of the unpredictability of a system gives a
measure of its complexity. Adopting this point of view, we review some
developments in the characterization of the predictability of systems showing
different kind of complexity: from low-dimensional systems to high-dimensional
ones with spatio-temporal chaos and to fully developed turbulence. A special
attention is devoted to finite-time and finite-resolution effects on
predictability, which can be accounted with suitable generalization of the
standard indicators. The problems involved in systems with intrinsic randomness
is discussed, with emphasis on the important problems of distinguishing chaos
from noise and of modeling the system. The characterization of irregular
behavior in systems with discrete phase space is also considered.Comment: 142 Latex pgs. 41 included eps figures, submitted to Physics Reports.
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