Manifold problems

• Manifold problems
• • Generic manifold problem
  • • Basic example
    • Projection function
    • Tangent function
    • Finalizing the solution

The idea behind MultiParamContinuation is to compute immersed manifolds $\mathcal M$ in memory limited environments, the manifold being defined as (parts) of the zeros of

\[F:\mathbb R^n\times \mathbb R^{par}\to\mathbb R^m.\]

Generic manifold problem

ManifoldProblem is the basic / generic structure for encoding a bifurcation problem ; it holds the following fields:

• the vector field
• an initial guess
• a set of parameters

as well as user defined functions for

• recording (record_from_solution) indicators about the solution when this one is too large to be saved at every continuation step.

and some other that we described below. A detailed account for the struct is ManifoldProblem.

Basic example

using MultiParamContinuation

F(u,p) = [u[3]]

prob = ManifoldProblem(F, [0,0,0.], nothing)

Projection function

You can pass your own projection function which takes a point $u_{guess}\in \mathbb R^n$ and project it on $\mathcal M$. For example, in the case of the plane F(u,p) = u[3], we could use:

prob = ManifoldProblem(F, [0,0,0.], nothing;
                        project = (u_g,p) -> begin
                        		u = copy(u_g)
                        		u[3] = 0
                        		return u
                        end,
                        )

Tangent function

You can pass your own tangent function which returns an orthonormal basis of the tangent space at point $u$. For example, in the case of the plane F(u,p) = u[3], we could use:

prob = ManifoldProblem(F, [0,0,0.], nothing;
                        get_tangent = (u,p) -> [1 0; 0 1; 0 0],
                        )

Finalizing the solution

You can pass a finalizer function which returns a boolean. This boolean is used to discard or not the current chart. This can be used to restrict the manifold to a given bounding box.

You can have a look at bounding space for predefined finalizer functions for simple spaces.

prob = ManifoldProblem(F, [.0,0.,0.], nothing;
                        finalize_solution = (c::Chart, p) -> true
                        )