Incorporate a Newton-Raphson Unstable Periodic Orbit numerical finder

[ushham]

Implement Newton-Raphson method for numerically finding unstable periodic orbits of continuous dynamical systems.

The linear and tensor correction method are outlined in this paper.
A. S. Gritsun, “Unstable periodic trajectories of a barotropic model of the atmosphere,” Russian Journal of Numerical Analysis and Mathematical Modelling 23 (2008), 10.1515/RJNAMM.2008.021.

A julia library exists that can numerically find UPOs using the Newton method: https://github.com/ushham/Jupo.jl. This library should be incorporated into ChaosTools.jl.

[Datseris]

Hi Oisin, nice to hear from you! We have already started a dedicated package for finding periodic orbits, stable or unstable. I've transferred the issue there. You can have a look https://juliadynamics.github.io/PeriodicOrbits.jl/dev/ and let me know if the API is good? We have not registered the package yet, I wanted to give more people the opportunity to have a look.

[Datseris]

Oh, by the way, this repo PeriodicOrbits.jl started so that we can implement a local continuation in DynamicalSystems.jl. I see in the Jupo.jl repo there is a file for continuation? If you are interested to chat about this as well let me know!

(But a key property of this repo is to on-purpose separate the finding orbits from the continuation part. I have concluded that because existing continuation frameworks conflate the two, this makes them so inaccessible for beginner contributors)

[ushham]

Hey George, long time no chat! This looks like a great base, and thank you for bringing this project to my attention.

Yes the base API looks great, and with a few modifications hopefully the code I have developed for the Newton method for continuous systems can be integrated into this project. I can provide more details on what would need changing in my codebase, and what would need to be added in this project, if you want.

I am a little busy at the moment but I hope to have some time later this summer and I can help move the code over.

On the continuation stuff, what I have implemented is brute force and dumb, and just slowly alters parameters and checks the resulting UPO is "close" to the original for certain characteristics, so probably not worth looking into.

[Datseris]

Sounds good, I will wait for your PR then, or comments on what needs changing!

On the continuation stuff, what I have implemented is brute force and dumb, and just slowly alters parameters and checks the resulting UPO is "close" to the original for certain characteristics, so probably not worth looking into.

That's how most continuation methods work actually haha! The function poequal can make this more robust though.