Features Overview

The features offered in this documentation section come from the package ChaosTools.jl. If you are encountering an issue with some of the methods, you can report/open a new issue at the GitHub Issues page.

Lyapunov Exponents

The following treat systems where the equations of motion are known:

  1. Maximum Lyapunov exponent for both discrete and continuous systems: lyapunov.
  2. Lyapunov spectrum for both discrete and continuous systems: lyapunovs.

Entropies and Dimensions

  1. Generalized (Renyi) entropy and all related entropies: genentropy.
  2. Very fast and very cheap (memory-wise) method for computing entropies of large datasets.
  3. Generalized dimensions (e.g. capacity dimension, information dimension, etc.): generalized_dim.
  4. Kaplan-Yorke dimension: kaplanyorke_dim.
  5. Automated detection of best algorithmic parameters for calculating attractor dimensions.

And, in order to automatically deduce dimensions, we also offer methods for:

  • Partitioning a function y(x) vs. x into regions where it is approximated by a straight line, using a flexible algorithm with a lot of control over the outcome. See linear_regions.
  • Detection of largest linear region of a function y(x) vs. x and extraction of the slope of this region.

Nonlinear Timeseries Analysis

  1. Flexible and abstracted Reconstruction interface, that creates the delay-coordinates reconstruction of a timeseries efficiently.
  2. Methods for estimating good Reconstruction parameters.

  3. Four different algorithms for numerically determining the maximum Lyapunov exponent of a (e.g. experimentally) measured timeseries: numericallyapunov.

    • Fast computation of the above algorithms made possible by combining the

    performance of NearestNeighbors.jl with the abstraction of ChaosTools.jl.

Periodicity

  1. Numerical method to find unstable and stable fixed points of any ordern of a discrete map (of any dimensionality): periodicorbits.

    • Convenience functions for defining and realizing all possible combinations of \mathbf{\Lambda}_k matrices required in the above method.

Chaos Detection

  1. The Generalized Alignment Index: \text{GALI}_k : gali.

    • Implemented for both discrete and continuous systems.