Bifurcation problems

• Bifurcation problems
• • Generic bifurcation problem
  • • Example
  • Problem modification
  • • Example
  • Minimal interface of AbstractBifurcationProblem
  • • Core (always required)
    • Required per functionality
    • Defaults and remarks
  • 🚧🚧 Automatic option setting (work in progress) 🚧🚧

The idea behind BifurcationKit is to compute bifurcation diagrams in memory limited environments where the device can barely hold the current continuation state. We thus disable by default saving all solutions along the branch and all eigenvectors (see ContinuationPar to change this behavior). Still, one needs to save a few solution indicators, like for plotting. This is the reason for the function record_from_solution (see below).

Generic bifurcation problem

BifurcationProblem 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
• a parameter axis

as well as user defined functions for

• plotting, plot_solution
• recording (record_from_solution) indicators about the solution when this one is too large to be saved at every continuation step.
• update function update!

Example

f(x,p) = @. sin(x * p.a)u0 = zeros(100_000_000) params = (a = 1.0, b = 2.0)# record a few components / indicators about x myRecord(x,p;k...) = (x1 = x[1], max = maximum(x), nrm = norm(x, Inf))prob = BifurcationProblem(f, u0, params, (@optic _.a);	record_from_solution = myRecord	)

Problem modification

In case you want to modify an existing problem, you should use the following method

re_make(
    prob;
    u0,
    params,
    lens,
    record_from_solution,
    plot_solution,
    J,
    Jᵗ,
    d2F,
    d3F
)

re_make(prob; u0 = new_u0)

Example

using BifurcationKitF(x,p) = @. p.a + x^2# parameterspar = (a = 0., b = 2)prob = BifurcationProblem(F, zeros(3), par, (@optic _.a))# change u0prob2 = BifurcationKit.re_make(prob, u0 = rand(3))

┌─ Bifurcation problem with uType Vector{Float64}
├─ Inplace: false
├─ Dimension: 3
├─ Symmetric: false
└─ Parameter: a

Minimal interface of AbstractBifurcationProblem

BifurcationProblem / ODEBifProblem already implement the whole interface. A custom type struct MyPB <: BifurcationKit.AbstractBifurcationProblem … must provide the methods below, depending on the functionalities it will be used with.

Core (always required)

• residual(pb, x, p) — the vector field $F(x,p)$
• jacobian(pb, x, p) — its derivative $J(x,p)=d_xF$
• getparams(pb), getlens(pb) — parameters and continuation axis (or the fields params, lens)
• getu0(pb) — initial guess (defaults to the field u0)

Required per functionality

Defaults and remarks

Most generic helpers assume the conventional fields u0, params, lens, recordFromSolution, plotSolution (and VF for the jet), see getu0, getparams, setparam, re_make, apply_jacobian in the source. If your type stores these data differently, override the corresponding accessors, as well as re_make (used by the Fold/Hopf MA formulations and codim-2 continuation).

There is no generic fallback for getdelta, is_symmetric, has_adjoint, isinplace, save_solution, update! and the jet methods on an arbitrary AbstractBifurcationProblem. They are provided for the built-in problems built on a BifFunction (subtypes of AbstractAllJetBifProblem) and for BifurcationProblem / ODEBifProblem. For a custom type, either implement them or wrap your vector field in a BifFunction / subtype AbstractAllJetBifProblem so that the jet (dF, d2F, d3F, …) and the associated traits are obtained automatically.

When using BifurcationProblem, the functions J, Jᵗ, d2F, d3F are optional: if not provided, BifFunction differentiates residual with ForwardDiff, so F must be smooth and compatible with ForwardDiff (for large-scale problems, provide an analytic J).

🚧🚧 Automatic option setting (work in progress) 🚧🚧

Setting the continuation options can be difficult for new comers. In the case of small ODE, we suggest to use ODEBifProblem instead of BifurcationProblem. Indeed, in this case, most continuation options will be set up automatically so that very good performance is achieved.

In the case of BifurcationProblem, the user has to set up these options manually. This can be useful for large scale problems where the specificities of the problem have to be used.