Use BVP interface to encode periodic orbits

[rveltz]

We still have two implementations for PO and BVP whereas PO are a sub case of BVP.

Section should be taken from src/periodicorbit/Sections.jl. However, it seems Collocation and Trapeze from src/periodicorbit/ do not rely on src/periodicorbit/Sections.jl. Perhaps, we should refactor this first.

• The phase condition for Shooting relies on SectionSS
  • refactor sections in Collocation and Trapeze
• The phase condition for Collocation implements the constraint ∫_0^T < u(t), ϕ'(t) > dt = 0 where ϕ is the periodic orbit at the previous continuation step.
  • we should try to remove xπ from Collocation, there are not used for the phase constraints. Create a SectionCollocation which stores ϕ and ∂ϕ
  • put sections in Sections.jl

It is perhaps a good idea to

At this stage, the current tests should pass, except StuartLandoCollocationDisc

struct DiscretizedPO{Tmodel, Tdisc, Tcache, Ts} <: AbstractDiscretizedPO
        d_bvp::DiscretizedBVP{Tmodel, Tdisc, Tcache}
        section::Ts
end

todo list

• forward getters get_time_interval, ...
• do not use hard coded reshape, but rely on get_time_slices
• the code for residual (collocation, shooting) should be factorized
• one should create mesh cache during the call to discretize
• add DiscretizedPO in DiscretizeBVP.jl
• POBifProblem to .... In the constructor, add update_section_every_step::UInt = 1,jacobian= DenseAnalytical()

At this stage, all tests should pass.

• we must update the phase constraint at "each" continuation step. Currently, this relies on update!(wrap::PeriodicOrbitFunctionalColl, iter, state). We thus must implement update!(wrap::DiscretizedPO, iter, state). However, we have an issue here. Currently BVPBifProblem has a field update! which is called as update!(prob, x, p) = prob.update!(x,p). But for DiscretizedPO, I prefered to rely on dispatch rather than storing the function. There are two ways to call update!(getprob(it), it, state).

  1. Either it is stored in the problem and used as such https://github.com/bifurcationkit/BifurcationKit.jl/blob/master/src/Problems.jl#L367 for BifurcationProblem.
  2. But for PeriodicOrbitFunctionalColl I chosed to rely on dispatch and so I directly implemented it https://github.com/bifurcationkit/BifurcationKit.jl/blob/master/src/periodicorbit/PeriodicOrbitCollocation.jl#L1368. I suggest to rename L1368 as update_periodic_orbit_functional_coll and pass it to POBifProblem

I would rather to create a singleton struct; UpdateFunctionForPOFunctional; end, pass it as en update! function and dispatch on it.

• put periodic orbits implementation in src/bvp
• remove folder src/periodicorbit

[rveltz]

To do the merge, I suggest to copy the test stuartLandauCollocation.jl which has been renamed to stuartLandauCollocationDisc.jl. One of those files must be removed at the end.