🟢 Sphere based on BifurcationKit.jl

In this tutorial, we show how to cover a sphere as solution of

\[F(u) := \|u\|^2-1 = 0\]

We use this model as a mean to introduce the basics of MultiParamContinuation.jl based on BifurcationKit.jl.

It is easy to encode the manifold using ManifoldProblem_BK which yields a ManifoldProblemBK with BifurcationKit internals.

using CairoMakie, BifurcationKit, MultiParamContinuation
const MPC = MultiParamContinuation

F(u,p) = [u[1]^2 + u[2]^2 + u[3]^2 - 1]

prob = ManifoldProblem_BK(F,
                    [1,0.,0.],
                    nothing
                        )

show(prob)

2-d Manifold Problem
    ├─ n = 3
    └─ m = 1

We compute a covering of the manifold:

S = MPC.continuation(prob,
            Henderson(np0 = 6),
            CoveringPar(max_charts = 20000,
                    max_steps = 250,
                    # trigger the use of BifurcationKit newton solver
                    newton_options = NewtonPar(),
                    R0 = .2,
                    )
            )
show(S)

Surface 2-d
   ├─ # charts = 230
   └─ problem =
          2-d Manifold Problem
              ├─ n = 3
              └─ m = 1

You can now plot the result as

MPC.plotd(S;
    draw_tangent = true,
    plot_center = false,
    draw_edges = true,
    ind_plot = [1,3])