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Titel |
The Geometry of Chaotic Dynamics - A Complex Network Perspective |
VerfasserIn |
Reik Donner, Jobst Heitzig, Jonathan F. Donges, Yong Zou, Norbert Marwan, Jürgen Kurths |
Konferenz |
EGU General Assembly 2011
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Medientyp |
Artikel
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Sprache |
Englisch
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Digitales Dokument |
PDF |
Erschienen |
In: GRA - Volume 13 (2011) |
Datensatznummer |
250056142
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Zusammenfassung |
Recently, several complex network approaches to time series analysis have been developed
and applied to study a wide range of model systems as well as real-world data, e.g.,
geophysical or financial time series. Among these techniques, recurrence-based concepts and
prominently É-recurrence networks, most faithfully represent the geometrical fine structure of
the attractors underlying chaotic (and less interestingly non-chaotic) time series. In this paper
we demonstrate that the well known graph theoretical properties local clustering coefficient
and global (network) transitivity can meaningfully be exploited to define two new local
and two new global measures of dimension in phase space: local upper and lower
clustering dimension as well as global upper and lower transitivity dimension. Rigorous
analytical as well as numerical results for self-similar sets and simple chaotic model
systems suggest that these measures are well-behaved in most non-pathological
situations and that they can be estimated reasonably well using É-recurrence networks
constructed from relatively short time series. Moreover, we study the relationship between
clustering and transitivity dimensions on the one hand, and traditional measures
like pointwise dimension or local Lyapunov dimension on the other hand. We also
provide further evidence that the local clustering coefficients, or equivalently the local
clustering dimensions, are useful for identifying unstable periodic orbits and other
dynamically invariant objects from time series. Our results demonstrate that É-recurrence
networks exhibit an important link between dynamical systems and graph theory. |
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