324 research outputs found
Editorial Comment on the Special Issue of "Information in Dynamical Systems and Complex Systems"
This special issue collects contributions from the participants of the
"Information in Dynamical Systems and Complex Systems" workshop, which cover a
wide range of important problems and new approaches that lie in the
intersection of information theory and dynamical systems. The contributions
include theoretical characterization and understanding of the different types
of information flow and causality in general stochastic processes, inference
and identification of coupling structure and parameters of system dynamics,
rigorous coarse-grain modeling of network dynamical systems, and exact
statistical testing of fundamental information-theoretic quantities such as the
mutual information. The collective efforts reported herein reflect a modern
perspective of the intimate connection between dynamical systems and
information flow, leading to the promise of better understanding and modeling
of natural complex systems and better/optimal design of engineering systems
<Contributed Talk 40>A combinatorial framework for nonlinear dynamics
[Date] November 28 (Mon) - December 2 (Fri), 2011: [Place] Kyoto University Clock Tower Centennial Hall, Kyoto, JAPA
<Contributed Talk 38>Topological-Computational Methods for Analyzing Global Dynamics and Bifurcations
[Date] November 28 (Mon) - December 2 (Fri), 2011: [Place] Kyoto University Clock Tower Centennial Hall, Kyoto, JAPA
Discretization strategies for computing Conley indices and Morse decompositions of flows
Conley indices and Morse decompositions of flows can be found by using
algorithms which rigorously analyze discrete dynamical systems. This usually
involves integrating a time discretization of the flow using interval
arithmetic. We compare the old idea of fixing a time step as a parameters to a
time step continuously varying in phase space. We present an example where this
second strategy necessarily yields better numerical outputs and prove that our
outputs yield a valid Morse decomposition of the given flow
Inducing a map on homology from a correspondence
We study the homomorphism induced in homology by a closed correspondence
between topological spaces, using projections from the graph of the
correspondence to its domain and codomain. We provide assumptions under which
the homomorphism induced by an outer approximation of a continuous map
coincides with the homomorphism induced in homology by the map. In contrast to
more classical results we do not require that the projection to the domain have
acyclic preimages. Moreover, we show that it is possible to retrieve correct
homological information from a correspondence even if some data is missing or
perturbed. Finally, we describe an application to combinatorial maps that are
either outer approximations of continuous maps or reconstructions of such maps
from a finite set of data points
Using machine learning to predict catastrophes in dynamical systems
Nonlinear dynamical systems, which include models of the Earth\u27s climate, financial markets and complex ecosystems, often undergo abrupt transitions that lead to radically different behavior. The ability to predict such qualitative and potentially disruptive changes is an important problem with far-reaching implications. Even with robust mathematical models, predicting such critical transitions prior to their occurrence is extremely difficult. In this work, we propose a machine learning method to study the parameter space of a complex system, where the dynamics is coarsely characterized using topological invariants. We show that by using a nearest neighbor algorithm to sample the parameter space in a specific manner, we are able to predict with high accuracy the locations of critical transitions in parameter space. (C) 2011 Elsevier B.V. All rights reserved
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