DSWeb Dynamical Systems Software aims to collect all available software on dynamical systems theory. This project was originally launched during the special year Emerging Applications of Dynamical Systems, 1997/1998, at the Institute for Mathematics and its Applications. The information here includes functionality, platforms, languages, references, and contacts.

Please note that DSWeb is not responsible for any direct, indirect, special, incidental, or consequential damages arising from the use of the content provided here.

# Classify Chimeras

### Runner-up - DSWeb 2018 Software Contest

By Felix P. Kemeth, Sindre W. Haugland, Ioannis G. Kevrekidis, and Katharina Krischer
Print Requirements

Following packages are required:

Installation

Via pip: (sudo) pip install classify_chimeras

Via source: https://github.com/fkemeth/classify_chimeras

Documentation

This python package contains functions to classify chimera states, non-linear hybrid states of coexisting coherence and incoherence. In partical, this package offers three functions, following the paper "A classification scheme for chimera states" (http://dx.doi.org/10.1063/1.4959804)

• spatial(A, boundaries='no-flux', phases=False, nbins=100) A must be a TxN or a TxN1xN2 numpy matrix (either real or complex). The function spatial() applies the discrete Laplacian on the data, and returns the coherent fraction at each time step. boundaries specifies the boundary conditions under which the data was generated. Set phases=True if A contains phases only. nbins specifies the number of bins of the histograms which are generated.
• globaldist(A, nbins=100, phases=False, Ncoarse=1500) A must be a TxN numpy matrix. The function globaldist() calculates all pariwise Euclidean distances between all data points at each time step, and returns the coherent fraction of A at each time step. nbins specifies the number of bins of the histograms. Set phases=True if A contains phases only. Ncoarse is a threshold above which the data is coarsed due to memory limitations. This can be increased, but may lead to long calculation times or memory errors.
• temporal(A, nbins=100, phases=False, Ncoarse=1500) A must be a TxN or TxN1xN2 numpy matrix. The function temporal() calculates all pairwise temporal correlation coefficients between the T-long timeseries of A. It returns a hisogram, with the square root of the last bin indicating the amount of temporarily correlated time series. nbins specifies the number of bins of the histograms. Set phases=True if A contains phases only. Ncoarse is a threshold above which the data is coarsed due to memory limitations. This can be increased, but may lead to long calculation times or memory errors.

Example

As an illustrative example, we use a chimera state observed by Kuramoto and Battogtokh in "Coexistence of Coherence and Incoherence in Nonlocally Coupled Phase Oscillators" (2002), in Nonlinear Phenom. Complex Syst. We suppose that we have the phases of this chimera state in a numpy matrix A.

import classify_chimeras as clc
import pylab as pl

# Plot a snapshot of the data matrix A
(T,N) = A.shape
pl.plot(np.arange(0,1,1.0/float(N)),A[-1],'.'); pl.show()

![Snapshot of the phases](/images/kuramoto.jpg)

# Obtain the fraction of spatially coherent oscillators
g0 = clc.spatial(A, boundaries='periodic', phases=True)
pl.plot(g0); pl.ylim((0,1.0)); pl.show()

![Fraction of spatially coherent oscillators](/images/kuramoto_g0.jpg)

# Obtain the fraction of temporarily correlated oscillators
h = clc.temporal(A, phases=True)
pl.plot(h); pl.ylim((0,0.3)); pl.show()
h0 = np.sqrt(h[-1])

![Distribution of temporal correlation coefficients](/images/kuramoto_h.jpg)

Licence

This work is licenced under GNU General Public License v3. This means you must cite "A classification scheme for chimera states" F.P. Kemeth et al. (http://dx.doi.org/10.1063/1.4959804) if you use this package for publications.

 Model Time Series Software Type Package Language Python Platform UnixLinuxWindowsMacOS Availability https://github.com/fkemeth/classify_chimeras Contact Person Felix P. Kemeth References to Papers F. P. Kemeth, S. W. Haugland, L. Schmidt, I. G. Kevrekidis, and K. Krischer, A classification scheme for chimera states, Chaos: An Interdisciplinary Journal of Nonlinear Science 26, 094815 (2016), https://doi.org/10.1063/1.4959804.