The logistic map is a typical example for dynamical transitions between regular, laminar, and chaotic behavior of a dynamical system. The evolution of the time series depends on the control parameter . The time series is defined by the iterative map .

The frequency distribution shows the frequency of occurrence of different values of the resulting time series.

Although the time series changes in the fixed range for different initial values , the frequency distribution (outside the fixed points) stays approximately the same.

For the plots we omit the first 1000 iterations in order to get rid of transient behavior in our analysis. However, to visualize the dynamics properly, we included only 100 time steps in the time series plot. For the frequency distribution we used 4000 time steps.

The snapshots show the phase-space density profiles of the logistic map for four different dynamical regimes (periodic, band-merging chaotic, laminar chaotic, fully chaotic). The frequency distribution is not sufficient to quantitatively distinguish these regimes nor to investigate the transitions between them.

This Demonstration was created by Alraune Zech during an internship at the Potsdam Institute for Climate Impact Research, Germany. It is based on the following 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, 2009 pp. 4246–4254.