`fixedpoints` based on root finding

[reykboerner]

Describe the feature you'd like to have
I'd like to suggest making the fixedpoints method from ChaosTools.jl available in Attractors.jl. This would allow finding attractors and edge states also via root finding, in addition to the current ways (via proximity, recurrence, featurizing). I would assume that for systems with fixed points this can be the most efficient way?

[Datseris]

I was actually thinking that this method should be ported to PeriodicOrbits.jl (which then itself would be used in Attractors.jl). However PeriodicOrbits.jl isn't finished yet.

the fixed point method fits much better the periodic Orbits.jl API: you get a periodic orbit of period 1 (dicrete time) or 0 (continuous time). PeriodicOrbits.jl is also for both stable and unstable orbits while Attractors.jl is only for stable. In contrast, the fixed points function doesn't fit the "mapper" infrastructure that Attractors.lj is based on.

But yeah the main bottlenect is that a small amount of polishing is still remaining for PeriodicOrbits.jl before it is publised.

[Datseris]

I would assume that for systems with fixed points this can be the most efficient way?

It depends on how many attractors there are in the system. For relatively few, the recurrences method is a super efficient way for fixed points, but it only works for stable ones.

Do note that our fixed point function uses interval arethmetic. So there is some sort of "guarantee" that it will find all fixed points, but I haven't actually tested how well the guarantee holds in real world use cases. It can get pretty expensive plus it requires a smooth enough Jacobian which many dynamial systems don't have.