This is a package devoted to different tools for analyzing (primarily 2D) 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
- The Birkhoff extrapolation method for finding invariant circles [1]
- The reproducing kernel Hilbert space based invariant level-set finding method [2]
- The implementations of the parameterization method for finding invariant circles and connecting orbits (see, e.g., [3] for an introduction to the parameterization method)
- 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 usingLiterate.jlout of files found indocs/, and can be recreated locally by runningdocs/literate_examples.jl. Additionally, these notebooks are outputted to markdown and included in the documentation.
[1] (Submitted) M. Ruth and D. Bindel, Finding Birkhoff Averages via Adaptive Filtering, Mar. 2024. arXiv:2403.19003
[math.DS].
[2] (Submitted) M. Ruth and D. Bindel, Level Set Learning for Poincaré Plots of Symplectic Maps, Dec. 2023. arXiv:2312.00967 [physics].
[3] 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.
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.
This is a new package, and things are changing rapidly! Currently, features involving the InvariantCircle struct, the extrapolation algorithm, and periodic orbits (1, 3, and 4 above) are mostly tested, with the features related to (3) still in need of testing. More examples are being added as well.
For any questions, email mer335@cornell.edu