Overview of BifurcationKit.jl

• Overview of BifurcationKit.jl
• • Defining Problems
  • Solving the problem
  • Analyzing the branch

The general workflow for using the package is as follows:

• Define a problem
• Solve the problem
• Analyze the output

Defining Problems

Problems are specified via a type interface. The problem types are designed to contain the necessary information to fully define their associated continuation method. For example, a bifurcation problem is defined by

\[F(u, pars) = 0\]

with some parameters pars, some initial guess u0, and scalar parameter axis lens contained in pars. Therefore, the BifurcationProblem is defined by those components:

prob = BifurcationProblem(F, u0, pars, lens)

Note that the number types in the solution will match the types you designate in the problem. However, complex types are not advised as they mess up the detection of bifurcation points.

Several problems are available, see Bifurcation problems for more details.

Solving the problem

Each type of problem has its own type which allows the solvers to dispatch to the right methods. The common interface for calling the solvers is:

br = continuation(prob, alg, contparams::ContinuationPar; kwargs)

Into the command, one passes the problem that was defined prob, chooses an algorithm alg, and changes the properties of the solver using keyword arguments. The solver returns a branch object br which holds all the details for the branch.

Several continuation algorithms are provided, see Continuation methods: introduction.

Analyzing the branch

The solution type has a common interface, which makes handling the solution similar between the different types of problems. Tools such as interpolations are seamlessly built into the solution interface to make analysis easy. This interface is described in the ContResult: it contains the continuation curve, the detected bifurcation points in br.specialpoint, the eigen elements used to assess stability, etc.

Plotting functionality is provided by a recipe to Plots.jl. To use plot branches, simply call the plot(br) and the plotter will generate appropriate plots. Plots can be customized using all the keyword arguments provided by Plots.jl. Please see Plots.jl's documentation for more information.