100 research outputs found
Kinetics of Fluid Demixing in Complex Plasmas: Domain Growth Analysis using Minkowski Tensors
A molecular dynamics simulation of the demixing process of a binary complex
plasma is analysed and the role of distinct interaction potentials is discussed
by using morphological Minkowski tensor analysis of the minority phase domain
growth in a demixing simulated binary complex plasma. These Minkowski tensor
methods are compared with previous results that utilized a power spectrum
method based on the time-dependent average structure factor. It is shown that
the Minkowski tensor methods are superior to the previously used power spectrum
method in the sense of higher sensitivity to changes in domain size. By
analysis of the slope of the temporal evolution of Minkowski tensor measures
qualitative differences between the case of particle interaction with a single
length scale compared to particle interactions with two different length scales
(dominating long range interaction) are revealed. After proper scaling the
graphs for the two length scale scenario coincide, pointing towards universal
behaviour. The qualitative difference in demixing scenarios is evidenced by
distinct demixing behaviour: In the long range dominated cases demixing occurs
in two stages. At first neighbouring particles agglomerate then domains start
to merge in cascades. However in the case of only one interaction length scale
only agglomeration but no merging of domains can be observed. Thus, Minkowski
Tensor analysis are likely to become a useful tool for further investigation of
this (and other) demixing processes. It is capable to reveal (nonlinear) local
topological properties, probing deeper than (linear) global power spectrum
analysis, however still providing easily interpretable results founded on a
solid mathematical framework.Comment: 12 pages, 10 figures, Phys. Rev. E, accepted for publication,
http://journals.aps.org/pr
Instability onset and scaling laws of an autooscillating turbulent flow in a complex plasma
We study a complex plasma under microgravity conditions that is first
stabilized with an oscillating electric field. Once the stabilization is
stopped, the so-called heartbeat instability develops. We study how the kinetic
energy spectrum changes during and after the onset of the instability and
compare with the double cascade predicted by Kraichnan and Leith for
two-dimensional turbulence. The onset of the instability manifests clearly in
the ratio of the reduced rates of cascade of energy and enstrophy and in the
power-law exponents of the energy spectra.Comment: 7 pages, 7 figure
Time series with tailored nonlinearities
It is demonstrated how to generate time series with tailored nonlinearities by inducing well-defined constraints
on the Fourier phases. Correlations between the phase information of adjacent phases and (static and dynamic)
measures of nonlinearities are established and their origin is explained. By applying a set of simple constraints on
the phases of an originally linear and uncorrelated Gaussian time series, the observed scaling behavior of the intensity distribution of empirical time series can be reproduced. The power law character of the intensity distributions being typical for, e.g., turbulence and financial data can thus be explained in terms of phase correlations
Linear and nonlinear market correlations: characterizing financial crises and portfolio optimization
Pearson correlation and mutual information based complex networks of the
day-to-day returns of US S&P500 stocks between 1985 and 2015 have been
constructed in order to investigate the mutual dependencies of the stocks and
their nature. We show that both networks detect qualitative differences
especially during (recent) turbulent market periods thus indicating strongly
fluctuating interconnections between the stocks of different companies in
changing economic environments. A measure for the strength of nonlinear
dependencies is derived using surrogate data and leads to interesting
observations during periods of financial market crises. In contrast to the
expectation that dependencies reduce mainly to linear correlations during
crises we show that (at least in the 2008 crisis) nonlinear effects are
significantly increasing. It turns out that the concept of centrality within a
network could potentially be used as some kind of an early warning indicator
for abnormal market behavior as we demonstrate with the example of the 2008
subprime mortgage crisis. Finally, we apply a Markowitz mean variance portfolio
optimization and integrate the measure of nonlinear dependencies to scale the
investment exposure. This leads to significant outperformance as compared to a
fully invested portfolio.Comment: 12 pages, 11 figures, Phys. Rev. E, accepte
Breaking Symmetries of the Reservoir Equations in Echo State Networks
Reservoir computing has repeatedly been shown to be extremely successful in
the prediction of nonlinear time-series. However, there is no complete
understanding of the proper design of a reservoir yet. We find that the
simplest popular setup has a harmful symmetry, which leads to the prediction of
what we call mirror-attractor. We prove this analytically. Similar problems can
arise in a general context, and we use them to explain the success or failure
of some designs. The symmetry is a direct consequence of the hyperbolic tangent
activation function. Further, four ways to break the symmetry are compared
numerically: A bias in the output, a shift in the input, a quadratic term in
the readout, and a mixture of even and odd activation functions. Firstly, we
test their susceptibility to the mirror-attractor. Secondly, we evaluate their
performance on the task of predicting Lorenz data with the mean shifted to
zero. The short-time prediction is measured with the forecast horizon while the
largest Lyapunov exponent and the correlation dimension are used to represent
the climate. Finally, the same analysis is repeated on a combined dataset of
the Lorenz attractor and the Halvorsen attractor, which we designed to reveal
potential problems with symmetry. We find that all methods except the output
bias are able to fully break the symmetry with input shift and quadratic
readout performing the best overall.Comment: 14 pages, 10 figures, accepted by chao
Controlling dynamical systems to complex target states using machine learning: next-generation vs. classical reservoir computing
Controlling nonlinear dynamical systems using machine learning allows to not
only drive systems into simple behavior like periodicity but also to more
complex arbitrary dynamics. For this, it is crucial that a machine learning
system can be trained to reproduce the target dynamics sufficiently well. On
the example of forcing a chaotic parametrization of the Lorenz system into
intermittent dynamics, we show first that classical reservoir computing excels
at this task. In a next step, we compare those results based on different
amounts of training data to an alternative setup, where next-generation
reservoir computing is used instead. It turns out that while delivering
comparable performance for usual amounts of training data, next-generation RC
significantly outperforms in situations where only very limited data is
available. This opens even further practical control applications in real world
problems where data is restricted.Comment: IJCNN 202
Synchronization in systems with linear, yet nonreciprocal interactions
Synchronization of oscillatory subsystems is a widespread phenomenon in science. It is argued that the presence of nonlinearities is a necessary prerequisite for synchronization. Here, we study synchronization in
complex plasmas consisting of microparticles in addition to the plasma. The particles can form 2D crystalline structures. They can melt via mode-coupling instability (MCI), which is a consequence of the effective nonreciprocal interactions. Synchronized particle motion during MCI-melting of 2D plasma crystal was reported in [1]. To disentangle the effects of nonlinearity and nonreciprocity on the emergence of synchronization,
we solved numerically the nonlinear and the linearized
system. Analyzing the synchronization with a new order parameter [2] reveals that a linearized version of the interaction model exhibits the same synchronization patterns as the full, nonlinear one. Further,
theoretical considerations show that nonreciprocal interactions among particles generally provide a mechanism for the selection of dominant
wave modes causing the system to show synchronized motion. Thus, we demonstrate numerically and analytically that also linear systems can synchronize and that the nonreciprocity of the interaction is the
more decisive property for a n-body system to synchronize.
[1] L. Couëdel et al., Phys. Rev. E, 89, 053108 (2014)
[2] I. Laut et al., EPL, 110, 65001 (2015
Reducing network size and improving prediction stability of reservoir computing
Reservoir computing is a very promising approach for the prediction of
complex nonlinear dynamical systems. Besides capturing the exact short-term
trajectories of nonlinear systems, it has also proved to reproduce its
characteristic long-term properties very accurately. However, predictions do
not always work equivalently well. It has been shown that both short- and
long-term predictions vary significantly among different random realizations of
the reservoir. In order to gain an understanding on when reservoir computing
works best, we investigate some differential properties of the respective
realization of the reservoir in a systematic way. We find that removing nodes
that correspond to the largest weights in the output regression matrix reduces
outliers and improves overall prediction quality. Moreover, this allows to
effectively reduce the network size and, therefore, increase computational
efficiency. In addition, we use a nonlinear scaling factor in the hyperbolic
tangent of the activation function. This adjusts the response of the activation
function to the range of values of the input variables of the nodes. As a
consequence, this reduces the number of outliers significantly and increases
both the short- and long-term prediction quality for the nonlinear systems
investigated in this study. Our results demonstrate that a large optimization
potential lies in the systematical refinement of the differential reservoir
properties for a given dataset.Comment: 11 pages, 8 figures, published in Chao
Calibrated reservoir computers
We observe the presence of infinitely fine-scaled alternations within the performance landscape of reservoir computers aimed for chaotic data forecasting. We investigate the emergence of the observed structures by means of variations of the transversal stability of the synchronization manifold relating the observational and internal dynamical states. Finally, we deduce a simple calibration method in order to attenuate the thus evidenced performance uncertainty
Controlling nonlinear dynamical systems into arbitrary states using machine learning
Controlling nonlinear dynamical systems is a central task in many different areas of science and
engineering. Chaotic systems can be stabilized (or chaotified) with small perturbations, yet existing
approaches either require knowledge about the underlying system equations or large data sets as they
rely on phase space methods. In this work we propose a novel and fully data driven scheme relying on
machine learning (ML), which generalizes control techniques of chaotic systems without requiring a
mathematical model for its dynamics. Exploiting recently developed ML-based prediction capabilities,
we demonstrate that nonlinear systems can be forced to stay in arbitrary dynamical target states
coming from any initial state. We outline and validate our approach using the examples of the Lorenz
and the Rössler system and show how these systems can very accurately be brought not only to
periodic, but even to intermittent and different chaotic behavior. Having this highly flexible control
scheme with little demands on the amount of required data on hand, we briefly discuss possible
applications ranging from engineering to medicine
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