sprint #09 / Use case KEEP: follow the limit cycle around an optimum

[abavoil]

In BifurcationKit.jl:

• define limit cycle boundary condition with either $g(x_0, x_f) = 0$ (more general) or $g(x_0) - g(x_f) = 0$ (limit cycle of a projected state). For the most general case, have $g$ also depend on time.

In KEEP.jl:

1. Make the source code public
2. Do I need to use (mutable) Arrays instead of StaticArrays? Probably not.
3. Is the use of ForwardDiff in my flow function a problem? Probably not, if I also use ForwardDiff in BifurcationKit.

[abavoil]

• $g(x_0, x_f, p)$ semble être le plus général.

• Où le code se trouve-t-il ? $x_0 = x_f$ défini ...

[abavoil]

ODE

ODE state: $x = (α, τ, \dot α, \dot τ, W)$
ODE parameters: $Θ$
ODE:
1: $\dot x = f(x, Θ)$
2: $x(0) = (α_0, 0, \dot α_0, \dot τ_0, 0)$

Optimization

Optimization state: Shooting $α_0, \dot α_0, \dot τ_0, W_f , t_f$ with parameters $Θ$
Optimization constraints:
3: $α(t_f, Θ) = α_0$
4: $τ(t_f, Θ) = τ_f = \pm 2π$ (and $τ(0) = 0$)
5: $\dot α(t_f, Θ) = \dot α$
6: $\dot τ(t_f, Θ) = \dot τ$
7: $W(t_f, Θ) = W_f$ (and $W(0) = 0$)
Maximized quantity: $W_f / t_f$

Where is the periodic orbit ?

For a given set of parameters $Θ$, we can determine $α_0, \dot α_0, \dot τ_0, t_f$ such that 3-6. These shooting parameters give a periodic orbit of our system in euclidean space: physically, the kite returned to its original position with its original velocity; but $τ(f, Θ) = τ_0 \pm 2π \neq τ_0$ and $W(f, Θ)$ can be whatever.
We give an initial guess for the optimization problem by solving the shooting problem and reading $W(t_f, Θ)$, thus it satisfies constraints 3-7.

[abavoil]

I should be able to solve this problem by simply redefining a few functions in my script, without changing the source code. Using a BVProblem instead of a periodic orbit ?

[rveltz]

We can write a specific for this problem in an example, based on BK internals, that we pack in a conventional BifurcationProblem. This will serve as a landmark before the API settles down and you would have a working example.