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Introduction & Contents

DynamicalSystems.jl logo: The Double Pendulum


DynamicalSystems.jl is an award-winning Julia software library for the exploration of chaos and nonlinear dynamics.

DSWeb Competition 1st place winner

DynamicalSystems.jl has won the first place in the 2018 Software Contest held by the dynamical systems division of SIAM. For more details see the official statement


DynamicalSystems.jl is part of JuliaDynamics, check out our website for more cool stuff!

Latest news

The 0-1 test for chaos is now included in DynamicalSystems.jl (testchaos01), as well as the Pomeau-Manneville map! Both results of a Hacktoberfest 2019 hackathon!

The documentation you are reading now was built with the following stable versions:

 * DelayEmbeddings ........... 1.3.0
 * RecurrenceAnalysis ........ 1.1.0
 * DynamicalSystemsBase ...... 1.5.0
 * ChaosTools ................ 1.8.3
 * InteractiveChaos .......... 0.3.2

Introductory textbooks

Our library assumes some basic knowledge of nonlinear dynamics and complex systems.

If you are new to the field but want to learn more, we can suggest the following textbooks as introductions:

  • Nonlinear Dynamics And Chaos - S. Strogatz
  • An Exploration of Dynamical Systems and Chaos - J. Argyris et al.
  • Chaos in Dynamical Systems - E. Ott
  • Nonlinear Time series Analysis - H. Kantz & T. Schreiber
  • Chaos and Integrability in Nonlinear Dynamics - M. Tabor

Jupyter Notebooks / Tutorials

In this repository you can find various Jupyter notebooks that have been used as introductory tutorials for DynamicalSystems.jl!

YouTube tutorial

You can find a tutorial on DynamicalSystems.jl hosted on the official YouTube channel of the Julia language:


The module DynamicalSystems re-exports all following functionality, grouped into different packages.


  • Intuitive, consistent APIs for the definition of general dynamical systems, under a unified struct DynamicalSystem. The following combinations are possible:

    • Continuous or Discrete systems. Continuous systems use DifferentialEquations.jl for solving the ODE problem.
    • In-place or out-of-place (large versus small systems).
    • Auto-differentiated or not (for the Jacobian function).
  • Automatic "completion" of the dynamics of the system with numerically computed Jacobians, in case they are not provided by the user.

  • Robust implementations of all kinds of integrators, that evolve the system, many states of the system, or even deviation vectors. See the advanced documentation for this.
  • Library of predefined well-known dynamical systems that have been used extensively in scientific research.


Is a package for performing delay coordinate embeddings and finding optimal parameters for doing so.


Is a package that has many algorithms for chaotic dynamical systems. All algorithms are independent of each other but they are also not expansive enough to be a standalone package.

Please see the overview section for a full list of features. Here is a quick summary:

  • Poincare S.O.S. and orbit diagrams
  • Lyapunov Exponents
  • Entropies and Dimensions
  • Lyapunov exponent of a timeseries (numerical data)
  • Periodic behavior
  • Categorizing and distinguishing chaotic and regular behavior
  • Nonlinear timeseries analysis


RecurrenceAnalysis offers tools to compute and analyze Recurrence Plots, a field called Recurrence Quantification Analysis.


InteractiveChaos is a package that provides applications for interactively exploring dynamical systems. It is an extension of DynamicalSystems.jl and builds upon existing code by hooking up to the DynamicalSystem structure.

InteractiveChaos is not installed with DynamicalSystems.jl. To install it do ]add InteractiveChaos Makie. Makie is necessary for providing a plotting backend, since InteractiveChaos does not install one by default.

The functionality of InteractiveChaos is contained within individual functions, all of which launch a dedicated interactive application. Here is their list:

Videos & Animations

Besides the documentation strings, each interactive function is accompanied with an animation (.gif or .mp4 file) displayed after the docstring, as well as a video tutorial demonstrating its use. See the individual pages for the video links (by clicking the documentation string links)!

Our Goals

The ultimate goal for DynamicalSystems.jl is to be a useful software library for students and scientists working on chaos, nonlinear dynamics and in general dynamical systems. The word "library" is intended in the literal sense: a place where people go to learn things.

With DynamicalSystems.jl we try to

  1. Be concise, intuitive, and general. All functions we offer work just as well with any system, whether it is a simple continuous chaotic system, like the Lorenz attractor, or a high dimensional discrete map like coupled standard maps.
  2. Be accurate, reliable and performant.
  3. Be transparent with respect to what is happening "under the hood", i.e. be clear about exactly what each function call does. We take care of this aspect in many ways; by being well-documented, giving references to scientific papers and having clear source code.


Simply use ]add DynamicalSystems to install everything. Alternatively you can also do using Pkg; Pkg.add("DynamicalSystems").

For more advanced users, you can choose which packages to install and use at a high level. The package DynamicalSystems serves two purposes: it re-exports everything under a single module DynamicalSystems and it also builds the documentation.

All packages depend on DelayEmbeddings which defines core numeric data structures and methods. For example RecurrenceAnalysis and TimeseriesPrediction depend only on DelayEmbeddings. Packages that require equations of motion also depend on DynamicalSystemsBase, like for example ChaosTools.

If you only need functionality of a specific package you can install only that one, e.g. ]add RecurrenceAnalysis and only the minimum amount of requirements will be installed.


There is a (very small) paper associated with DynamicalSystems.jl. If we have helped you in research that led to a publication, please be kind enough to cite it, using the DOI 10.21105/joss.00598 or the following BiBTeX entry:

  doi = {10.21105/joss.00598},
  url = {},
  year  = {2018},
  month = {mar},
  volume = {3},
  number = {23},
  pages = {598},
  author = {George Datseris},
  title = {DynamicalSystems.jl: A Julia software library for chaos and nonlinear dynamics},
  journal = {Journal of Open Source Software}

Issues with Bounties

Money that DynamicalSystems.jl obtains from awards, sponsors or donators are converted into bounties for GitHub issues. The full list of issues that have a bounty is available here.

By solving these issues you not only contribute to open source, but you also get some pocket money to boot :)


You can join our chatroom for discussions and/or questions about the packages of the JuliaDynamics organization! If you are using the Julia Slack workplace, please join the channel #dynamics-bridged.

Contributing & Donating

Be sure to visit the Contributor Guide page, because you can help make this package better without having to write a single line of code! Also, if you find this package helpful please consider staring it on GitHub! This gives us an accurate lower bound of users that this package has already helped!

Finally, you can donate for the development of DynamicalSystems.jl. You can do that by adding bounties to existing issues on the GitHub repositories (you can open new issues as well). Every issue has an automatic way to create a bounty using Bountysource, see the first comment of each issue.