8,294 research outputs found
Comparative Study of Homotopy Analysis and Renormalization Group Methods on Rayleigh and Van der Pol Equations
A comparative study of the Homotopy Analysis method and an improved
Renormalization Group method is presented in the context of the Rayleigh and
the Van der Pol equations. Efficient approximate formulae as functions of the
nonlinearity parameter for the amplitudes of the
limit cycles for both these oscillators are derived. The improvement in the
Renormalization group analysis is achieved by invoking the idea of nonlinear
time that should have significance in a nonlinear system. Good approximate
plots of limit cycles of the concerned oscillators are also presented within
this framework.Comment: 25 pages, 7 figures. Revised and upgraded: Differ Equ Dyn Syst, (26
July, 2015
Persistent Chaos in High Dimensions
An extensive statistical survey of universal approximators shows that as the
dimension of a typical dissipative dynamical system is increased, the number of
positive Lyapunov exponents increases monotonically and the number of parameter
windows with periodic behavior decreases. A subset of parameter space remains
in which topological change induced by small parameter variation is very
common. It turns out, however, that if the system's dimension is sufficiently
high, this inevitable, and expected, topological change is never catastrophic,
in the sense chaotic behavior is preserved. One concludes that deterministic
chaos is persistent in high dimensions.Comment: 4 pages, 3 figures; Changes in response to referee comment
Hysteresis in Adiabatic Dynamical Systems: an Introduction
We give a nontechnical description of the behaviour of dynamical systems
governed by two distinct time scales. We discuss in particular memory effects,
such as bifurcation delay and hysteresis, and comment the scaling behaviour of
hysteresis cycles. These properties are illustrated on a few simple examples.Comment: 28 pages, 10 ps figures, AMS-LaTeX. This is the introduction of my
Ph.D. dissertation, available at
http://dpwww.epfl.ch/instituts/ipt/berglund/these.htm
Delayed Dynamical Systems: Networks, Chimeras and Reservoir Computing
We present a systematic approach to reveal the correspondence between time
delay dynamics and networks of coupled oscillators. After early demonstrations
of the usefulness of spatio-temporal representations of time-delay system
dynamics, extensive research on optoelectronic feedback loops has revealed
their immense potential for realizing complex system dynamics such as chimeras
in rings of coupled oscillators and applications to reservoir computing.
Delayed dynamical systems have been enriched in recent years through the
application of digital signal processing techniques. Very recently, we have
showed that one can significantly extend the capabilities and implement
networks with arbitrary topologies through the use of field programmable gate
arrays (FPGAs). This architecture allows the design of appropriate filters and
multiple time delays which greatly extend the possibilities for exploring
synchronization patterns in arbitrary topological networks. This has enabled us
to explore complex dynamics on networks with nodes that can be perfectly
identical, introduce parameter heterogeneities and multiple time delays, as
well as change network topologies to control the formation and evolution of
patterns of synchrony
Beyond the Cosmological Standard Model
After a decade and a half of research motivated by the accelerating universe,
theory and experiment have a reached a certain level of maturity. The
development of theoretical models beyond \Lambda, or smooth dark energy, often
called modified gravity, has led to broader insights into a path forward, and a
host of observational and experimental tests have been developed. In this
review we present the current state of the field and describe a framework for
anticipating developments in the next decade. We identify the guiding
principles for rigorous and consistent modifications of the standard model, and
discuss the prospects for empirical tests. We begin by reviewing attempts to
consistently modify Einstein gravity in the infrared, focusing on the notion
that additional degrees of freedom introduced by the modification must screen
themselves from local tests of gravity. We categorize screening mechanisms into
three broad classes: mechanisms which become active in regions of high
Newtonian potential, those in which first derivatives become important, and
those for which second derivatives are important. Examples of the first class,
such as f(R) gravity, employ the familiar chameleon or symmetron mechanisms,
whereas examples of the last class are galileon and massive gravity theories,
employing the Vainshtein mechanism. In each case, we describe the theories as
effective theories. We describe experimental tests, summarizing laboratory and
solar system tests and describing in some detail astrophysical and cosmological
tests. We discuss future tests which will be sensitive to different signatures
of new physics in the gravitational sector. Parts that are more relevant to
theorists vs. observers/experimentalists are clearly indicated, in the hope
that this will serve as a useful reference for both audiences, as well as
helping those interested in bridging the gap between them.Comment: 175 pages, 24 figures. v2: Minor corrections, added references.
Review article, comments welcom
The Parameter Houlihan: a solution to high-throughput identifiability indeterminacy for brutally ill-posed problems
One way to interject knowledge into clinically impactful forecasting is to
use data assimilation, a nonlinear regression that projects data onto a
mechanistic physiologic model, instead of a set of functions, such as neural
networks. Such regressions have an advantage of being useful with particularly
sparse, non-stationary clinical data. However, physiological models are often
nonlinear and can have many parameters, leading to potential problems with
parameter identifiability, or the ability to find a unique set of parameters
that minimize forecasting error. The identifiability problems can be minimized
or eliminated by reducing the number of parameters estimated, but reducing the
number of estimated parameters also reduces the flexibility of the model and
hence increases forecasting error. We propose a method, the parameter Houlihan,
that combines traditional machine learning techniques with data assimilation,
to select the right set of model parameters to minimize forecasting error while
reducing identifiability problems. The method worked well: the data
assimilation-based glucose forecasts and estimates for our cohort using the
Houlihan-selected parameter sets generally also minimize forecasting errors
compared to other parameter selection methods such as by-hand parameter
selection. Nevertheless, the forecast with the lowest forecast error does not
always accurately represent physiology, but further advancements of the
algorithm provide a path for improving physiologic fidelity as well. Our hope
is that this methodology represents a first step toward combining machine
learning with data assimilation and provides a lower-threshold entry point for
using data assimilation with clinical data by helping select the right
parameters to estimate
Fluctuations in Nonequilibrium Statistical Mechanics: Models, Mathematical Theory, Physical Mechanisms
The fluctuations in nonequilibrium systems are under intense theoretical and
experimental investigation. Topical ``fluctuation relations'' describe
symmetries of the statistical properties of certain observables, in a variety
of models and phenomena. They have been derived in deterministic and, later, in
stochastic frameworks. Other results first obtained for stochastic processes,
and later considered in deterministic dynamics, describe the temporal evolution
of fluctuations. The field has grown beyond expectation: research works and
different perspectives are proposed at an ever faster pace. Indeed,
understanding fluctuations is important for the emerging theory of
nonequilibrium phenomena, as well as for applications, such as those of
nanotechnological and biophysical interest. However, the links among the
different approaches and the limitations of these approaches are not fully
understood. We focus on these issues, providing: a) analysis of the theoretical
models; b) discussion of the rigorous mathematical results; c) identification
of the physical mechanisms underlying the validity of the theoretical
predictions, for a wide range of phenomena.Comment: 44 pages, 2 figures. To appear in Nonlinearity (2007
Emerging unitary evolutions in dissipatively coupled systems
Having a broad range of methods available for implementing unitary operations is crucial for quantum information tasks. We study a dissipative process commonly used to describe dissipatively coupled systems and show that the process can lead to pure unitary dynamics on one part of a bipartite system, provided that the process is strong enough. As a consequence of these findings, we discuss within the framework of quantum control theory how the dissipative process can enable universal control of the considered part, thereby turning parts of the system into a system capable of universal quantum information tasks. We characterize the time scales necessary to implement gates with high fidelity through the dissipative evolution. The considered dissipative evolution is of particular importance since it can be engineered in the laboratory in the realm of superconducting circuits. Based on a reservoir that is formed by a lossy microwave mode we present a detailed study of how our theoretical findings can be realized in an experimental setting
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