# Introduction

DynamicalSystems.jl is a Julia software library for the exploration of chaos and nonlinear dynamics.

The current documentation was built with the following versions

 * DynamicalSystemsBase ...... 1.0.1
* ChaosTools ................ 1.1.2
* TimeseriesPrediction ...... 0.5.2


See the News page for recent updates. The latest one is the Julia 0.7 version support!

Introductory textbooks

Our library assumes basic 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

Jupyter Notebooks / Tutorials

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

## Video tutorial

The below tutorial is hosted on the Julia channel. Be careful though! It is using an older version of both DynamicalSystems.jl as well as Julia! Fortunately, from the side of DynamicalSystems.jl there have been very few breaking changes. See the News for more.

## Contents

### Fundamentals

1. Intuitive, consistent APIs for the definition of general dynamical systems, both maps and flows. 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).
2. Automatic "completion" of the dynamics of the system with numerically computed Jacobians, in case they are not provided by the user.

3. 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.
4. Dedicated interface for numerical data.
5. Efficient neighborhood estimation by interfacing NearestNeighbors.
6. Delay Coordinates Embedding: flexible and abstracted reconstruct interface, that creates the delay-coordinates reconstruction of a timeseries efficiently.

• Supports multiple dimensions and multiple timescales.
7. Library of predefined well-known dynamical systems that have been used extensively in scientific research.

### ChaosTools

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
• Estimation of Reconstruction parameters
• Lyapunov exponent of a timeseries
• Finding Fixed Points of Maps
• Detecting Chaos

### TimeseriesPrediction

Please see the introduction page for a detailed description!

• Predicting the future of one or multiple timeseries using average local models.
• Spatio-temporal timeseries prediction and cross-prediction.
• Multiple spatio-temporal embeddings.

## Our Goals

The ultimate goal for DynamicalSystems.jl is to be a useful library for students and scientists working on chaos, nonlinear dynamics and in general dynamical systems. We don't want to have "just code", but also detailed descriptions and references for as many methods as possible.

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.

## Installation

Simply use Pkg.add("DynamicalSystems") to install everything.

## Citing

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:

@article{Datseris2018,
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}
}