64,661 research outputs found
Recurrence networks - A novel paradigm for nonlinear time series analysis
This paper presents a new approach for analysing structural properties of
time series from complex systems. Starting from the concept of recurrences in
phase space, the recurrence matrix of a time series is interpreted as the
adjacency matrix of an associated complex network which links different points
in time if the evolution of the considered states is very similar. A critical
comparison of these recurrence networks with similar existing techniques is
presented, revealing strong conceptual benefits of the new approach which can
be considered as a unifying framework for transforming time series into complex
networks that also includes other methods as special cases.
It is demonstrated that there are fundamental relationships between the
topological properties of recurrence networks and the statistical properties of
the phase space density of the underlying dynamical system. Hence, the network
description yields new quantitative characteristics of the dynamical complexity
of a time series, which substantially complement existing measures of
recurrence quantification analysis
Spin Network States in Gauge Theory
Given a real-analytic manifold M, a compact connected Lie group G and a
principal G-bundle P -> M, there is a canonical `generalized measure' on the
space A/G of smooth connections on P modulo gauge transformations. This allows
one to define a Hilbert space L^2(A/G). Here we construct a set of vectors
spanning L^2(A/G). These vectors are described in terms of `spin networks':
graphs phi embedded in M, with oriented edges labelled by irreducible unitary
representations of G, and with vertices labelled by intertwining operators from
the tensor product of representations labelling the incoming edges to the
tensor product of representations labelling the outgoing edges. We also
describe an orthonormal basis of spin networks associated to any fixed graph
phi. We conclude with a discussion of spin networks in the loop representation
of quantum gravity, and give a category-theoretic interpretation of the spin
network states.Comment: 19 pages, LaTe
Bifurcation analysis in an associative memory model
We previously reported the chaos induced by the frustration of interaction in
a non-monotonic sequential associative memory model, and showed the chaotic
behaviors at absolute zero. We have now analyzed bifurcation in a stochastic
system, namely a finite-temperature model of the non-monotonic sequential
associative memory model. We derived order-parameter equations from the
stochastic microscopic equations. Two-parameter bifurcation diagrams obtained
from those equations show the coexistence of attractors, which do not appear at
absolute zero, and the disappearance of chaos due to the temperature effect.Comment: 19 page
Integrable cluster dynamics of directed networks and pentagram maps
The pentagram map was introduced by R. Schwartz more than 20 years ago. In
2009, V. Ovsienko, R. Schwartz and S. Tabachnikov established Liouville
complete integrability of this discrete dynamical system. In 2011, M. Glick
interpreted the pentagram map as a sequence of cluster transformations
associated with a special quiver. Using compatible Poisson structures in
cluster algebras and Poisson geometry of directed networks on surfaces, we
generalize Glick's construction to include the pentagram map into a family of
discrete integrable maps and we give these maps geometric interpretations. This
paper expands on our research announcement arXiv:1110.0472Comment: 46 pages, 22 figure
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