24,818 research outputs found
On the Collective Motion in Globally Coupled Chaotic Systems
A mean-field formulation is used to investigate the bifurcation diagram for
globally coupled tent maps by means of an analytical approach. It is shown that
the period doubling sequence of the single site map induces a continuous family
of periodic states in the coupled system. This type of collective motion breaks
the ergodicity of the coupled map lattice. The stability analysis suggests that
these states are stable for weak coupling strength but opens the possibility
for more complicated types of motion in the regime of moderate coupling.Comment: 12 pages, Latex, 3 eps figures included also available "at
http://athene.fkp.physik.th-darmstadt.de/public/wolfram.html" or "at
ftp://athene.fkp.physik.th-darmstadt.de/pub/publications/wolfram/" Phys. Rep.
in pres
The Time Invariance Principle, Ecological (Non)Chaos, and A Fundamental Pitfall of Discrete Modeling
This paper is to show that most discrete models used for population dynamics
in ecology are inherently pathological that their predications cannot be
independently verified by experiments because they violate a fundamental
principle of physics. The result is used to tackle an on-going controversy
regarding ecological chaos. Another implication of the result is that all
continuous dynamical systems must be modeled by differential equations. As a
result it suggests that researches based on discrete modeling must be closely
scrutinized and the teaching of calculus and differential equations must be
emphasized for students of biology
From order to chaos in Earth satellite orbits
We consider Earth satellite orbits in the range of semi-major axes where the
perturbing effects of Earth's oblateness and lunisolar gravity are of
comparable order. This range covers the medium-Earth orbits (MEO) of the Global
Navigation Satellite Systems and the geosynchronous orbits (GEO) of the
communication satellites. We recall a secular and quadrupolar model, based on
the Milankovitch vector formulation of perturbation theory, which governs the
long-term orbital evolution subject to the predominant gravitational
interactions. We study the global dynamics of this two-and-a-half
degrees-of-freedom Hamiltonian system by means of the fast Lyapunov indicator
(FLI), used in a statistical sense. Specifically, we characterize the degree of
chaoticity of the action space using angle-averaged normalized FLI maps,
thereby overcoming the angle dependencies of the conventional stability maps.
Emphasis is placed upon the phase-space structures near secular resonances,
which are of first importance to the space debris community. We confirm and
quantify the transition from order to chaos in MEO, stemming from the critical
inclinations, and find that highly inclined GEO orbits are particularly
unstable. Despite their reputed normality, Earth satellite orbits can possess
an extraordinarily rich spectrum of dynamical behaviors, and, from a
mathematical perspective, have all the complications that make them very
interesting candidates for testing the modern tools of chaos theory.Comment: 30 pages, 9 figures. Accepted for publication in the Astronomical
Journa
SOM-VAE: Interpretable Discrete Representation Learning on Time Series
High-dimensional time series are common in many domains. Since human
cognition is not optimized to work well in high-dimensional spaces, these areas
could benefit from interpretable low-dimensional representations. However, most
representation learning algorithms for time series data are difficult to
interpret. This is due to non-intuitive mappings from data features to salient
properties of the representation and non-smoothness over time. To address this
problem, we propose a new representation learning framework building on ideas
from interpretable discrete dimensionality reduction and deep generative
modeling. This framework allows us to learn discrete representations of time
series, which give rise to smooth and interpretable embeddings with superior
clustering performance. We introduce a new way to overcome the
non-differentiability in discrete representation learning and present a
gradient-based version of the traditional self-organizing map algorithm that is
more performant than the original. Furthermore, to allow for a probabilistic
interpretation of our method, we integrate a Markov model in the representation
space. This model uncovers the temporal transition structure, improves
clustering performance even further and provides additional explanatory
insights as well as a natural representation of uncertainty. We evaluate our
model in terms of clustering performance and interpretability on static
(Fashion-)MNIST data, a time series of linearly interpolated (Fashion-)MNIST
images, a chaotic Lorenz attractor system with two macro states, as well as on
a challenging real world medical time series application on the eICU data set.
Our learned representations compare favorably with competitor methods and
facilitate downstream tasks on the real world data.Comment: Accepted for publication at the Seventh International Conference on
Learning Representations (ICLR 2019
Modal decomposition of fluid-structure interaction with application to flag flapping
Modal decompositions such as proper orthogonal decomposition (POD), dynamic
mode decomposition (DMD) and their variants are regularly used to educe
physical mechanisms of nonlinear flow phenomena that cannot be easily
understood through direct inspection. In fluid-structure interaction (FSI)
systems, fluid motion is coupled to vibration and/or deformation of an immersed
structure. Despite this coupling, data analysis is often performed using only
fluid or structure variables, rather than incorporating both. This approach
does not provide information about the manner in which fluid and structure
modes are correlated. We present a framework for performing POD and DMD where
the fluid and structure are treated together. As part of this framework, we
introduce a physically meaningful norm for FSI systems. We first use this
combined fluid-structure formulation to identify correlated flow features and
structural motions in limit-cycle flag flapping. We then investigate the
transition from limit-cycle flapping to chaotic flapping, which can be
initiated by increasing the flag mass. Our modal decomposition reveals that at
the onset of chaos, the dominant flapping motion increases in amplitude and
leads to a bluff-body wake instability. This new bluff-body mode interacts
triadically with the dominant flapping motion to produce flapping at the
non-integer harmonic frequencies previously reported by Connell & Yue (2007).
While our formulation is presented for POD and DMD, there are natural
extensions to other data-analysis techniques
Chaotic Diffusion in the Gliese-876 Planetary System
Chaotic diffusion is supposed to be responsible for orbital instabilities in
planetary systems after the dissipation of the protoplanetary disk, and a
natural consequence of irregular motion. In this paper we show that resonant
multi-planetary systems, despite being highly chaotic, not necessarily exhibit
significant diffusion in phase space, and may still survive virtually unchanged
over timescales comparable to their age.Using the GJ-876 system as an example,
we analyze the chaotic diffusion of the outermost (and less massive) planet. We
construct a set of stability maps in the surrounding regions of the Laplace
resonance. We numerically integrate ensembles of close initial conditions,
compute Poincar\'e maps and estimate the chaotic diffusion present in this
system. Our results show that, the Laplace resonance contains two different
regions: an inner domain characterized by low chaoticity and slow diffusion,
and an outer one displaying larger values of dynamical indicators. In the outer
resonant domain, the stochastic borders of the Laplace resonance seem to
prevent the complete destruction of the system. We characterize the diffusion
for small ensembles along the parameters of the outermost planet. Finally, we
perform a stability analysis of the inherent chaotic, albeit stable Laplace
resonance, by linking the behavior of the resonant variables of the
configurations to the different sub-structures inside the three-body resonance.Comment: 13 pages, 7 figures, 2 tables. Accepted for publication in MNRA
Nonlinear Time-Domain Structure/Aerodynamics Coupling in Systems with Concentrated Structural Nonlinearities
This paper details a practical approach for predicting the aeroelastic response (structure/aerodynamics coupling) of flexible pod/missile-type configurations with freeplay/hysteresis concentrated structural nonlinearities. The nonlinear aeroelastic response of systems in the presence of these nonlinearities has been previously studied by different authors; this paper compiles methodologies and related airworthiness regulations. The aeroelastic equations of the pod/missile configuration are formulated in state-space form and time-domain integrated with Fortran/Matlab codes developed ad hoc for dealing with freeplay/hysteresis nonlinearities. Results show that structural nonlinearities change the classical aeroelastic behaviour with appearence of non-damped motion (LCOs and chaotic motion)
Energetics, skeletal dynamics and long-term predictions in Kolmogorov-Lorenz systems
We study a particular return map for a class of low dimensional chaotic
models called Kolmogorov Lorenz systems, which received an elegant general
Hamiltonian description and includes also the famous Lorenz63 case, from the
viewpoint of energy and Casimir balance. In particular it is considered in
detail a subclass of these models, precisely those obtained from the Lorenz63
by a small perturbation on the standard parameters, which includes for example
the forced Lorenz case in Ref.[6]. The paper is divided into two parts. In the
first part the extremes of the mentioned state functions are considered, which
define an invariant manifold, used to construct an appropriate Poincare surface
for our return map. From the experimental observation of the simple orbital
motion around the two unstable fixed points, together with the circumstance
that these orbits are classified by their energy or Casimir maximum, we
construct a conceptually simple skeletal dynamics valid within our sub class,
reproducing quite well the Lorenz map for Casimir. This energetic approach
sheds some light on the physical mechanism underlying regime transitions. The
second part of the paper is devoted to the investigation of a new type of
maximum energy based long term predictions, by which the knowledge of a
particular maximum energy shell amounts to the knowledge of the future
(qualitative) behaviour of the system. It is shown that, in this respect, a
local analysis of predictability is not appropriate for a complete
characterization of this behaviour. A perspective on the possible extensions of
this type of predictability analysis to more realistic cases in (geo)fluid
dynamics is discussed at the end of the paper.Comment: 21 pages, 14 figure
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