Recurrence Network Measures for the Logistic Map

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The degree centrality and clustering coefficient of nodes in the recurrence network of a time series reveal complementary geometrical properties of the dynamics in phase space. This helps to distinguish between the various dynamical regimes of a complex nonlinear system.

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The logistic map provides a good model for dynamical transitions among regular, laminar, and chaotic behavior of a dynamical system. The evolution and properties of the time series depend on the control parameter .

Observe how the degree and clustering series and frequency distribution evolve as you change the control parameter . Can you identify periodic windows in a sea of chaos? As you increase the recurrence threshold , the recurrence network measures slowly approach those expected for a fully connected network.

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Contributed by: Alraune Zech, Jonathan F. Donges, Norbert Marwan, and Jürgen Kurths (January 2010)
Open content licensed under CC BY-NC-SA


Snapshots


Details

We omit the first 1000 iterations in order to get rid of transient behavior in our analysis. To visualize the dynamics properly, we plot only the following 100 time steps in the upper panel.

Recurrence networks provide a complex network representation of the properties of a time series. Single observations are identified with nodes, where two nodes and are linked if and only if their mutual distance in phase space is smaller than a threshold value . Recurrence networks allow the application of measures from complex network theory to sophisticated analysis of time series and the study of general dynamical systems.

The degree centrality gives the number of neighbors of each node , . Hence it may be considered a measure of the local phase space density.

The local clustering coefficient gives the probability that two neighbors of the state are also neighbors: . The coefficient can be used to identify invariant objects in phase space, for example, supertrack functions in the bifurcation diagram of the logistic map or unstable periodic orbits (UPOs) in the phase space of continuous dynamical systems like the Rössler or Lorenz systems.

The snapshots show the degree series and distribution for various dynamical regimes of the logistic map (periodic, band merging chaotic, laminar chaotic, fully chaotic). The degree distributions differ substantially among the different regimes.

This Demonstration was created during an internship at the Potsdam Institute for Climate Impact Research, Germany. It is based on the articles:

N. Marwan, J. F. Donges, Y. Zou, R. V. Donner, and J. Kurths, "Complex Network Approach for Recurrence Analysis of Time Series," Physics Letters A, 373(46), 2009 pp. 4246–4254.

R. V. Donner, Y. Zou, J. F. Donges, N. Marwan, and J. Kurths, "Recurrence Networks - A Novel Paradigm for Nonlinear Time Series Analysis," arXiv, 2009.

R. V. Donner, Y. Zou, J. F. Donges, N. Marwan, and J. Kurths, "Ambiguities in Recurrence-Based Complex Network Representations of Time Series," Physical Review E (R), 2010 (in press).

Further information can be found at Recurrence Plots and Cross Recurrence Plots.



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