DynamicalSystems.jl logo: The Double Pendulum

DynamicalSystems.jl is an award-winning Julia software library for dynamical systems, nonlinear dynamics, deterministic chaos, and nonlinear timeseries analysis. It is part of JuliaDynamics, an organization dedicated to creating high quality scientific software.

To learn how to use this library please see Getting started below, and subsequently, the Contents page to get an overview of all offered functionality of DynamicalSystems.jl.

Latest news

Welcome to this new (and slightly breaking) release of DynamicalSystems.jl! You can find an announcement of the new version on the Julia discourse:


The breaking changes in this release are:

  1. The keyword dt of many functions has been renamed to Δt. This keyword had conflicts with the options of DifferentialEquations.jl. No warning can be thrown for this change, and users still using dt will have it silently propagated as keyword to the diffeq solvers. Functions affected: trajectory, lyapunov, lyapunovspectrum, gali, expansionentropy, orbitdiagram
  2. If A is a Dataset then A[range_of_integers] now returns a Dataset. Before it used to return Vector{SVector}.
Star us on GitHub!

If you have found this library useful, please consider starring it on GitHub. This gives us an accurate lower bound of the (satisfied) user count.

Getting started

DynamicalSystems.jl is a collection of Julia packages bundled together under a single package DynamicalSystems. To install this bundle you can do:

using Pkg; Pkg.add("DynamicalSystems")

The individual packages that compose DynamicalSystems interact flawlessly with each other because of the following two structures:

  1. The DynamicalSystem represents a dynamical system with known dynamic rule $f$. The system can be in discrete time (often called a map), $\vec{u}_{n+1} = \vec{f}(\vec{u}_n, p, n)$, or in continuous time (often called an ordinary differential equation) $\frac{d\vec{u}}{dt} = \vec{f}(\vec{u}, p, t)$. In both cases $u$ is the state of the dynamical system and $p$ a parameter container. You should have a look at the page Dynamical System Definition for how to create this object. A list of several pre-defined systems exists in the Predefined Dynamical Systems page.
  2. Numerical data, that can represent measured experiments, sampled trajectories of dynamical systems, or just sets in the state space, are represented by Dataset, which is a container of equally-sized data points. Timeseries in DynamicalSystems.jl are represented by the already existing Vector type of the Julia language.

These core structures DynamicalSystem, Dataset are used throughout the package to do useful calculations often used in the field of nonlinear dynamics and chaos. For example, using lyapunovspectrum and DynamicalSystem gives you the Lyapunov exponents of a dynamical system with known equations of motion. Alternatively, by using lyapunov_from_data and Dataset you can approximate the maximum Lyapunov exponent of a measured trajectory or a reconstructed set resulting from embed.

All things possible in DynamicalSystems.jl are listed in the Contents page. For transparency, the packages and versions used to build the documentation you are reading now are:

using Pkg
    "DelayEmbeddings", "RecurrenceAnalysis",
    "DynamicalSystemsBase", "ChaosTools",
    mode = PKGMODE_MANIFEST, io=stdout
      Status `~/work/DynamicalSystems.jl/DynamicalSystems.jl/docs/Manifest.toml`
  [608a59af] ChaosTools v2.5.0
  [5732040d] DelayEmbeddings v2.0.1
  [6e36e845] DynamicalSystemsBase v2.3.2
  [ed8fcbec] Entropies v1.1.0
  [639c3291] RecurrenceAnalysis v1.6.1


Tutorials for DynamicalSystems.jl exist in the form of Jupyter notebooks.

In addition, a full 2-hours YouTube tutorial is available below:

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:

  • Chaos in Dynamical Systems - E. Ott
  • Nonlinear Time series Analysis - H. Kantz & T. Schreiber

Advanced Installation

Notice that for targeted usage of DynamicalSystems (e.g. you only need a specific function like lyapunovspectrum or rqa), you don't have to install the entire DynamicalSystems suite. You can leave with only installing the necessary package that exports the function you need. You see this information prefacing the function. E.g. for rqa you will see RecurrenceAnalysis.rqa, which means that you need to install RecurrenceAnalysis to use it.

Our Goals

DynamicalSystems.jl was created with three goals in mind. The first was to fill the missing gap of a high quality and general purpose software for nonlinear dynamics. The second was to create a useful library where students and scientists from different fields may come and learn about methods of nonlinear dynamics.

The third was to fundamentally change the perception of the role of code in both scientific education as well as research. It is rarely the case that real, runnable code is shown in the classroom, because it is often long and messy. This is especially hurtful for nonlinear dynamics, a field where computer-assisted exploration is critical. But published work in this field fares even worse, with the overwhelming majority of published research not sharing the code used to create the paper. This makes reproducing these papers difficult, while some times straight-out impossible.

To achieve these goals we made DynamicalSystems.jl so that it is:

  1. Transparent: extra care is taken so that the source code of all functions is clear and easy to follow, while remaining as small and concise as possible.
  2. Intuitive: a software simple to use and understand makes experimentation easier.
  3. Easy to install, easy to extend: This makes contributions more likely, and can motivate researchers to implement their method here, instead of leaving it in a cryptic script stored in some data server, never-to-be-published with the paper.
  4. Reliable: the algorithm implementations are tested extensively.
  5. Well-documented: all implemented algorithms provide a high-level scientific description of their functionality in their documentation string as well as references to scientific papers.
  6. General: all algorithms work just as well with any system, whether it is a simple continuous chaotic system, like the Lorenz model, or a high dimensional discrete system like coupled standard maps.
  7. Performant: written entirely in Julia, and taking advantage of some of the best packages within the language, DynamicalSystems.jl is really fast.


There is a (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 = {https://doi.org/10.21105/joss.00598},
  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 :)

Asking questions

There are three options for asking questions:

  1. Join the official Julia discourse and ask a question under the category Specific Domains > Modelling & Simulations.
  2. Join our chatroom on gitter or the channel #dynamics-bridged in the Julia Slack workplace.
  3. Open an issue directly on the GitHub page of DynamicalSystems.jl while providing a Minimal Working Example.

Contributing & Donating

TL;DR: See "good first issues" or "wanted features".

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.

Maintainers and Contributors

The DynamicalSystems.jl software is maintained by George Datseris, who is also curating and writing this documentation page.

The software code however is built from the contributions of several individuals. For an accurate list of the names as well as contributions of each one, please visit our webpage.