SymplecticMapTools.jl

SymplecticMapTools is a package devoted to different tools for analyzing symplectic maps. The overall goal of this package is to provide a set of tools that can be used to robustly characterize orbits and find invariant structures while being efficient in the number of evaluations of the map. If you have any algorithms that you would like to see added, pull requests are welcome!

The implemented algorithms in this package include

1. The Birkhoff extrapolation method for finding invariant tori [1,2]
2. The reproducing kernel Hilbert space based invariant level-set finding method [3]
3. The implementations of the parameterization method for finding invariant circles and connecting orbits (see, e.g., [4] for an introduction to the parameterization method)
4. Newton's method with line search and BFGS for finding periodic orbits

Examples of how to use the code are found in examples/. These examples are created using Literate.jl out of files found in docs/, and can be recreated locally by running docs/literate_examples.jl. Additionally, these notebooks are outputted to markdown and included in the documentation.

[1] M. Ruth and D. Bindel, Finding Birkhoff Averages via Adaptive Filtering, Chaos 34, 123109 (2024) [Journal] [arXiv]
[2] M. Ruth, J. Kulik, and J. Burby, Robust computation of higher-dimensional invariant tori from individual trajectories, arXiv:2505.08715 [math.DS] [arXiv]
[3] M. Ruth and D. Bindel, Level Set Learning for Poincaré Plots of Symplectic Maps, SIAM Journal on Applied Dynamical Systems, Vol. 24, Iss. 1 (2025) [Journal] [arXiv]
[4] A. Haro, M. Canadell, J.-L. Figueras, A. Luque, and J. M. Mondelo, The Parameterization Method for Invariant Manifolds: From Rigorous Results to Effective Computations, vol. 195 of Applied Mathematical Sciences, Springer International Publishing, Cham, 2016.

Citation

To cite this package, we recommend you cite [1] or [2] for Birkhoff RRE, or [3] for the level set method.

Installation

For installation, add the package from the general repository using the command ]add SymplecticMapTools.jl in the Julia REPL. This package uses Requires for plotting. You must load CairoMakie or Plots separately to use the plotting functionality.

Examples

• Birkhoff Averaging
• Extrapolation Example
• Kernel Method for Finding Invariant Circles

Documentation

• Documentation
• • Contents
  • Index
  • Periodic Orbits
  • Invariant Circles
  • Fourier Circles
  • Fourier Tori
  • Connecting Orbits
  • Kernel Labels
  • Continued Fractions
  • Birkhoff Extrapolation
  • Lyapunov Exponents
  • Example Maps
  • Plotting Routines: Plots
  • Plotting Routines: CairoMakie
• Internal Documentation
• • Contents
  • Index
  • Fourier Circles
  • Birkhoff Extrapolation

Contact

For any questions, email maximilian.ruth@austin.utexas.edu.