42,704 research outputs found
Pseudo-nonstationarity in the scaling exponents of finite-interval time series
The accurate estimation of scaling exponents is central in the observational study of scale-invariant phenomena. Natural systems unavoidably provide observations over restricted intervals; consequently, a stationary stochastic process (time series) can yield anomalous time variation in the scaling exponents, suggestive of nonstationarity. The variance in the estimates of scaling exponents computed from an interval of N observations is known for finite variance processes to vary as ~1/N as N for certain statistical estimators; however, the convergence to this behavior will depend on the details of the process, and may be slow. We study the variation in the scaling of second-order moments of the time-series increments with N for a variety of synthetic and “real world” time series, and we find that in particular for heavy tailed processes, for realizable N, one is far from this ~1/N limiting behavior. We propose a semiempirical estimate for the minimum N needed to make a meaningful estimate of the scaling exponents for model stochastic processes and compare these with some “real world” time series
Complex Dynamics and Synchronization of Delayed-Feedback Nonlinear Oscillators
We describe a flexible and modular delayed-feedback nonlinear oscillator that
is capable of generating a wide range of dynamical behaviours, from periodic
oscillations to high-dimensional chaos. The oscillator uses electrooptic
modulation and fibre-optic transmission, with feedback and filtering
implemented through real-time digital-signal processing. We consider two such
oscillators that are coupled to one another, and we identify the conditions
under which they will synchronize. By examining the rates of divergence or
convergence between two coupled oscillators, we quantify the maximum Lyapunov
exponents or transverse Lyapunov exponents of the system, and we present an
experimental method to determine these rates that does not require a
mathematical model of the system. Finally, we demonstrate a new adaptive
control method that keeps two oscillators synchronized even when the coupling
between them is changing unpredictably.Comment: 24 pages, 13 figures. To appear in Phil. Trans. R. Soc. A (special
theme issue to accompany 2009 International Workshop on Delayed Complex
Systems
A New Test for Chaos
We describe a new test for determining whether a given deterministic
dynamical system is chaotic or nonchaotic. (This is an alternative to the usual
approach of computing the largest Lyapunov exponent.) Our method is a 0-1 test
for chaos (the output is a 0 signifying nonchaotic or a 1 signifying chaotic)
and is independent of the dimension of the dynamical system. Moreover, the
underlying equations need not be known. The test works equally well for
continuous and discrete time. We give examples for an ordinary differential
equation, a partial differential equation and for a map.Comment: 10 pages, 5 figure
Evidence of Intermittent Cascades from Discrete Hierarchical Dissipation in Turbulence
We present the results of a search of log-periodic corrections to scaling in
the moments of the energy dissipation rate in experiments at high Reynolds
number (2500) of three-dimensional fully developed turbulence. A simple
dynamical representation of the Richardson-Kolmogorov cartoon of a cascade
shows that standard averaging techniques erase by their very construction the
possible existence of log-periodic corrections to scaling associated with a
discrete hierarchy. To remedy this drawback, we introduce a novel ``canonical''
averaging that we test extensively on synthetic examples constructed to mimick
the interplay between a weak log-periodic component and rather strong
multiplicative and phase noises. Our extensive tests confirm the remarkable
observation of statistically significant log-periodic corrections to scaling,
with a prefered scaling ratio for length scales compatible with the value gamma
= 2. A strong confirmation of this result is provided by the identification of
up to 5 harmonics of the fundamental log-periodic undulations, associated with
up to 5 levels of the underlying hierarchical dynamical structure. A natural
interpretation of our results is that the Richardson-Kolmogorov mental picture
of a cascade becomes a realistic description if one allows for intermittent
births and deaths of discrete cascades at varying scales.Comment: Latex document of 40 pages, including 18 eps figure
The effect of round-off error on long memory processes
We study how the round-off (or discretization) error changes the statistical
properties of a Gaussian long memory process. We show that the autocovariance
and the spectral density of the discretized process are asymptotically rescaled
by a factor smaller than one, and we compute exactly this scaling factor.
Consequently, we find that the discretized process is also long memory with the
same Hurst exponent as the original process. We consider the properties of two
estimators of the Hurst exponent, namely the local Whittle (LW) estimator and
the Detrended Fluctuation Analysis (DFA). By using analytical considerations
and numerical simulations we show that, in presence of round-off error, both
estimators are severely negatively biased in finite samples. Under regularity
conditions we prove that the LW estimator applied to discretized processes is
consistent and asymptotically normal. Moreover, we compute the asymptotic
properties of the DFA for a generic (i.e. non Gaussian) long memory process and
we apply the result to discretized processes.Comment: 44 pages, 4 figures, 4 table
Coined quantum walks on percolation graphs
Quantum walks, both discrete (coined) and continuous time, form the basis of
several quantum algorithms and have been used to model processes such as
transport in spin chains and quantum chemistry. The enhanced spreading and
mixing properties of quantum walks compared with their classical counterparts
have been well-studied on regular structures and also shown to be sensitive to
defects and imperfections in the lattice. As a simple example of a disordered
system, we consider percolation lattices, in which edges or sites are randomly
missing, interrupting the progress of the quantum walk. We use numerical
simulation to study the properties of coined quantum walks on these percolation
lattices in one and two dimensions. In one dimension (the line) we introduce a
simple notion of quantum tunneling and determine how this affects the
properties of the quantum walk as it spreads. On two-dimensional percolation
lattices, we show how the spreading rate varies from linear in the number of
steps down to zero, as the percolation probability decreases to the critical
point. This provides an example of fractional scaling in quantum walk dynamics.Comment: 25 pages, 14 figures; v2 expanded and improved presentation after
referee comments, added extra figur
Orbital instability and relaxation in stellar systems
The orbits of stars in galaxies are generically chaotic: the chaotic behavior
arises in part from the intrinsically grainy nature of a potential that is
composed of point masses. Even if the potential is assumed to be smooth,
however, orbits in non-axisymmetric galaxies can be chaotic due to the presence
of central density cusps or black holes. The chaotic nature of orbits implies
that perturbations will grow exponentially and this in turn is expected to
result in a diffusion in phase space. We show that the degree of orbital
evolution is not well predicted by the growth rate of infinitesimal
perturbations, i.e. by the Liapunov exponent. A more useful criterion is
whether perturbations continue to grow exponentially until their scale is of
order the size of the system. We illustrate these ideas in a potential
consisting of N fixed point masses. Liapunov exponents are large for all values
of N, but orbits become increasingly regular in their behavior as N increases;
the reason is that the exponential divergence saturates at smaller and smaller
distances as N is increased. The objects which impede diffusion are the
invariant tori. When there are few stable tori, an ensemble of chaotic orbits
evolves rapidly toward a nearly stationary state. This mixing process occurs on
timescales of a few crossing times in triaxial potentials containing massive
central singularities, consistent with the rapid evolution observed in N-body
simulations of galaxies with central black holes.Comment: Invited review to appear in "The Chaotic Universe", eds. R. Ruffini,
V.G. Gurzadyan (World Scientific) 1999, 9 pages, 4 figure
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