360 research outputs found
Directional complexity and entropy for lift mappings
We introduce and study the notion of a directional complexity and entropy for
maps of degree 1 on the circle. For piecewise affine Markov maps we use
symbolic dynamics to relate this complexity to the symbolic complexity. We
apply a combinatorial machinery to obtain exact formulas for the directional
entropy, to find the maximal directional entropy, and to show that it equals
the topological entropy of the map. Keywords: Rotation interval, Space-time
window, Directional complexity, Directional entropy;Comment: 19p. 3 fig, Discrete and Continuous Dynamical Systems-B (Vol. 20, No.
10) December 201
On the statistical distribution of first--return times of balls and cylinders in chaotic systems
We study returns in dynamical systems: when a set of points, initially
populating a prescribed region, swarms around phase space according to a
deterministic rule of motion, we say that the return of the set occurs at the
earliest moment when one of these points comes back to the original region. We
describe the statistical distribution of these "first--return times" in various
settings: when phase space is composed of sequences of symbols from a finite
alphabet (with application for instance to biological problems) and when phase
space is a one and a two-dimensional manifold. Specifically, we consider
Bernoulli shifts, expanding maps of the interval and linear automorphisms of
the two dimensional torus. We derive relations linking these statistics with
Renyi entropies and Lyapunov exponents.Comment: submitted to Int. J. Bifurcations and Chao
Existence and Stability of Steady Fronts in Bistable CML
We prove the existence and we study the stability of the kink-like fixed
points in a simple Coupled Map Lattice for which the local dynamics has two
stable fixed points. The condition for the existence allows us to define a
critical value of the coupling parameter where a (multi) generalized
saddle-node bifurcation occurs and destroys these solutions. An extension of
the results to other CML's in the same class is also displayed. Finally, we
emphasize the property of spatial chaos for small coupling.Comment: 18 pages, uuencoded PostScript file, J. Stat. Phys. (In press
Detectability of non-differentiable generalized synchrony
Generalized synchronization of chaos is a type of cooperative behavior in
directionally-coupled oscillators that is characterized by existence of stable
and persistent functional dependence of response trajectories from the chaotic
trajectory of driving oscillator. In many practical cases this function is
non-differentiable and has a very complex shape. The generalized synchrony in
such cases seems to be undetectable, and only the cases, in which a
differentiable synchronization function exists, are considered to make sense in
practice. We show that this viewpoint is not always correct and the
non-differentiable generalized synchrony can be revealed in many practical
cases. Conditions for detection of generalized synchrony are derived
analytically, and illustrated numerically with a simple example of
non-differentiable generalized synchronization.Comment: 8 pages, 8 figures, submitted to PR
Adaptive coupling for achieving stable synchronization of chaos
We consider synchronization of coupled chaotic systems and propose an
adaptive strategy that aims at evolving the strength of the coupling to achieve
stability of the synchronized evolution. We test this idea in a simple
configuration in which two chaotic systems are unidirectionally coupled (a
sender and a receiver) and we study conditions for the receiver to adaptively
synchronize with the sender. Numerical simulations show that, under certain
conditions, our strategy is successful in dynamically evolving the coupling
strength until it converges to a value that is compatible with synchronization.Comment: 12 Pages, 9 figures, accepted for publication in Physical Review
The shock-acoustic waves generated by earthquakes
We investigate the form and dynamics of shock-acoustic waves generated by
earthquakes. We use the method for detecting and locating the sources of
ionospheric impulsive disturbances, based on using data from a global network
of receivers of the GPS navigation system and requiring no a priori information
about the place and time of associated effects. The practical implementation of
the method is illustrated by a case study of earthquake effects in Turkey
(August 17, and November 12, 1999), in Southern Sumatera (June 4, 2000), and
off the coast of Central America (January 13, 2001). It was found that in all
instances the time period of the ionospheric response is 180-390 s, and the
amplitude exceeds by a factor of two as a minimum the standard deviation of
background fluctuations in total electron content in this range of periods
under quiet and moderate geomagnetic conditions. The elevation of the wave
vector varies through a range of 20-44 degree, and the phase velocity
(1100-1300 m/s) approaches the sound velocity at the heights of the ionospheric
F-region maximum. The calculated (by neglecting refraction corrections)
location of the source roughly corresponds to the earthquake epicenter. Our
data are consistent with the present views that shock-acoustic waves are caused
by a piston-like movement of the Earth surface in the zone of an earthquake
epicenter.Comment: EmTeX-386, 30 pages, 4 figures, 3 tabl
Superdiffusion in the Dissipative Standard Map
We consider transport properties of the chaotic (strange) attractor along
unfolded trajectories of the dissipative standard map. It is shown that the
diffusion process is normal except of the cases when a control parameter is
close to some special values that correspond to the ballistic mode dynamics.
Diffusion near the related crisises is anomalous and non-uniform in time: there
are large time intervals during which the transport is normal or ballistic, or
even superballistic. The anomalous superdiffusion seems to be caused by
stickiness of trajectories to a non-chaotic and nowhere dense invariant Cantor
set that plays a similar role as cantori in Hamiltonian chaos. We provide a
numerical example of such a sticky set. Distribution function on the sticky set
almost coincides with the distribution function (SRB measure) of the chaotic
attractor.Comment: 10 Figure
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