Power Spectrum of the Logistic Map

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The power spectrum shows the energy distribution over different frequencies of a time series.

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

Although the time series changes for different initial values with fixed control parameter , the power spectrum (outside the fixed points) stays approximately the same.

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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

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 plotted the first 100 time steps in the time series plot. The power spectrum is calculated using the discrete Fourier transformation of the time series and 500 time steps.

The snapshots show the power spectrum of the logistic map time series for four different dynamical regimes (periodic, band merging chaotic, laminar chaotic, fully chaotic). The power spectrum is not sufficient to quantitatively distinguish these regimes or 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 [1] and [2].

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

References

[1] 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.

[2] R. V. Donner, Y. Zou, J. F. Donges, N. Marwan, and J. Kurths, "Recurrence Network—A Novel Paradigm for Nonlinear Time Series Analysis," 2009.



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