439 research outputs found
Output functions and fractal dimensions in dynamical systems
We present a novel method for the calculation of the fractal dimension of
boundaries in dynamical systems, which is in many cases many orders of
magnitude more efficient than the uncertainty method. We call it the Output
Function Evaluation (OFE) method. The OFE method is based on an efficient
scheme for computing output functions, such as the escape time, on a
one-dimensional portion of the phase space. We show analytically that the OFE
method is much more efficient than the uncertainty method for boundaries with
, where is the dimension of the intersection of the boundary with a
one-dimensional manifold. We apply the OFE method to a scattering system, and
compare it to the uncertainty method. We use the OFE method to study the
behavior of the fractal dimension as the system's dynamics undergoes a
topological transition.Comment: Uses REVTEX; to be published in Phys. Rev. Let
Variation of the Dependence of the Transient Process Duration on the Initial Conditions in Systems with Discrete Time
Dependence of the transient process duration on the initial conditions is
considered in one- and two-dimensional systems with discrete time, representing
a logistic map and the Eno map, respectively.Comment: 4 pages, 2 figure
Analog to Digital Conversion in Physical Measurements
There exist measuring devices where an analog input is converted into a
digital output. Such converters can have a nonlinear internal dynamics. We show
how measurements with such converting devices can be understood using concepts
from symbolic dynamics. Our approach is based on a nonlinear one-to-one mapping
between the analog input and the digital output of the device. We analyze the
Bernoulli shift and the tent map which are realized in specific analog/digital
converters. Furthermore, we discuss the sources of errors that are inevitable
in physical realizations of such systems and suggest methods for error
reduction.Comment: 9 pages in LATEX, 4 figures in ps.; submitted to 'Chaos, Solitons &
Fractals
Multistability and nonsmooth bifurcations in the quasiperiodically forced circle map
It is well-known that the dynamics of the Arnold circle map is phase-locked
in regions of the parameter space called Arnold tongues. If the map is
invertible, the only possible dynamics is either quasiperiodic motion, or
phase-locked behavior with a unique attracting periodic orbit. Under the
influence of quasiperiodic forcing the dynamics of the map changes
dramatically. Inside the Arnold tongues open regions of multistability exist,
and the parameter dependency of the dynamics becomes rather complex. This paper
discusses the bifurcation structure inside the Arnold tongue with zero rotation
number and includes a study of nonsmooth bifurcations that happen for large
nonlinearity in the region with strange nonchaotic attractors.Comment: 25 pages, 22 colored figures in reduced quality, submitted to Int. J.
of Bifurcation and Chaos, a supplementary website
(http://www.mpipks-dresden.mpg.de/eprint/jwiersig/0004003/) is provide
Dynamics towards the Feigenbaum attractor
We expose at a previously unknown level of detail the features of the
dynamics of trajectories that either evolve towards the Feigenbaum attractor or
are captured by its matching repellor. Amongst these features are the
following: i) The set of preimages of the attractor and of the repellor are
embedded (dense) into each other. ii) The preimage layout is obtained as the
limiting form of the rank structure of the fractal boundaries between attractor
and repellor positions for the family of supercycle attractors. iii) The joint
set of preimages for each case form an infinite number of families of
well-defined phase-space gaps in the attractor or in the repellor. iv) The gaps
in each of these families can be ordered with decreasing width in accord to
power laws and are seen to appear sequentially in the dynamics generated by
uniform distributions of initial conditions. v) The power law with log-periodic
modulation associated to the rate of approach of trajectories towards the
attractor (and to the repellor) is explained in terms of the progression of gap
formation. vi) The relationship between the law of rate of convergence to the
attractor and the inexhaustible hierarchy feature of the preimage structure is
elucidated.Comment: 8 pages, 12 figure
Collective Almost Synchronization in Complex Networks
This work introduces the phenomenon of Collective Almost Synchronization
(CAS), which describes a universal way of how patterns can appear in complex
networks even for small coupling strengths. The CAS phenomenon appears due to
the existence of an approximately constant local mean field and is
characterized by having nodes with trajectories evolving around periodic stable
orbits. Common notion based on statistical knowledge would lead one to
interpret the appearance of a local constant mean field as a consequence of the
fact that the behavior of each node is not correlated to the behaviors of the
others. Contrary to this common notion, we show that various well known weaker
forms of synchronization (almost, time-lag, phase synchronization, and
generalized synchronization) appear as a result of the onset of an almost
constant local mean field. If the memory is formed in a brain by minimising the
coupling strength among neurons and maximising the number of possible patterns,
then the CAS phenomenon is a plausible explanation for it.Comment: 3 figure
Emerging attractors and the transition from dissipative to conservative dynamics
The topological structure of basin boundaries plays a fundamental role in the
sensitivity to the initial conditions in chaotic dynamical systems. Herewith we
present a study on the dynamics of dissipative systems close to the Hamiltonian
limit, emphasising the increasing number of periodic attractors and on the
structural changes in their basin boundaries as the dissipation approaches
zero. We show numerically that a power law with nontrivial exponent describes
the growth of the total number of periodic attractors as the damping is
decreased. We also establish that for small scales the dynamics is governed by
\emph{effective} dynamical invariants, whose measure depends not only on the
region of the phase space, but also on the scale under consideration.
Therefore, our results show that the concept of effective invariants is also
relevant for dissipative systems.Comment: 9 pages, 10 figures. Accepted and in press for PR
Periodicity of chaotic trajectories in realizations of finite computer precisions and its implication in chaos communications
Fundamental problems of periodicity and transient process to periodicity of
chaotic trajectories in computer realization with finite computation precision
is investigated by taking single and coupled Logistic maps as examples.
Empirical power law relations of the period and transient iterations with the
computation precisions and the sizes of coupled systems are obtained. For each
computation we always find, by randomly choosing initial conditions, a single
dominant periodic trajectory which is realized with major portion of
probability. These understandings are useful for possible applications of
chaos, e.g., chaotic cryptography in secure communication.Comment: 10 pages, 3 figures, 2 table
Chaotic features in classical scattering processes between ions and atoms
A numerical study has been done of collisions between protons and hydrogen
atoms, treated as classical particles, at low impact velocities. The presence
of chaos has been looked for by investigating the processes with standard
techniques of the chaotic--scattering theory. The evidence of a sharp
transition from nearly regular scattering to fully developed chaos has been
found at the lower velocities.Comment: 10 pages, Latex, 3 figures (available upon request to the authors),
submitted to Journal of Physics
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