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
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:
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).