# The geometry of chaotic dynamics — a complex network perspective

@article{Donner2011TheGO, title={The geometry of chaotic dynamics — a complex network perspective}, author={Reik V. Donner and Jobst Heitzig and Jonathan F. Donges and Yong Zou and Norbert Marwan and J. Kurths}, journal={The European Physical Journal B}, year={2011}, volume={84}, pages={653-672} }

Abstract
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 ε-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…

## 139 Citations

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- 2016

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## References

SHOWING 1-10 OF 92 REFERENCES

### Recurrence networks—a novel paradigm for nonlinear time series analysis

- Computer Science
- 2010

It has been demonstrated here that there are fundamental relationships between many topological properties of recurrence networks and different nontrivial statistical properties of the phase space density of the underlying dynamical system.

### Complex network from pseudoperiodic time series: topology versus dynamics.

- Computer SciencePhysical review letters
- 2006

Standard measures of structure in complex networks can be applied to distinguish different dynamic regimes in time series and application to human electrocardiograms shows that such statistical properties are able to differentiate between the sinus rhythm cardiograms of healthy volunteers and those of coronary care patients.

### Analysis of Chaotic Dynamics Using Measures of the Complex Network Theory

- Computer ScienceICANN
- 2008

Deterministic chaos from a new aspect is analyzed, based on the idea that attractors of nonlinear dynamical systems and networks are characterized by a two-dimensional matrix: a recurrence plot and an adjacent matrix, and the networks constructed from the chaotic systems show a small world property.

### Complex network from time series based on phase space reconstruction.

- Computer ScienceChaos
- 2009

A reliable method for constructing complex networks from a time series with each vector point of the reconstructed phase space represented by a single node and edge determined by the phase space distance is proposed.

### Identifying complex periodic windows in continuous-time dynamical systems using recurrence-based methods.

- Computer ScienceChaos
- 2010

It is demonstrated that nonlinear measures based on recurrence plots obtained from such trajectories provide a practicable alternative for numerically detecting shrimps and the average path length and the clustering coefficient of the resulting recurrence networks are particularly powerful discriminatory statistics for the identification of complex periodic windows.

### Collective dynamics of ‘small-world’ networks

- Computer ScienceNature
- 1998

Simple models of networks that can be tuned through this middle ground: regular networks ‘rewired’ to introduce increasing amounts of disorder are explored, finding that these systems can be highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs.

### On the properties of small-world network models

- Physics
- 1999

Abstract:We study the small-world networks recently introduced by Watts and Strogatz [Nature 393, 440 (1998)], using analytical as well as numerical tools. We characterize the geometrical properties…

### From time series to complex networks: The visibility graph

- Computer Science, MathematicsProceedings of the National Academy of Sciences
- 2008

A simple and fast computational method, the visibility algorithm, that converts a time series into a graph, which inherits several properties of the series in its structure, enhancing the fact that power law degree distributions are related to fractality.