How can I save a solution every n steps, or at specific parameter values?

You can use the callback finalise_solution in the function call continuation. For example, you can use something like this to save all steps

function mySave(u, tau, step, contResult, personaldata)
	push!(personaldata, u)

and pass it like continuation(prob, alg, opts; finalise_solution = (z, tau, step, contResult; k...) -> mySave(z, tau, step, contResult, myData))

The Fold / Hopf Continuation does not work, why?

This requires some precise computations. Have you tried passing the expression of the Jacobian instead of relying on finite differences.

What is the parameter theta about in ContinuationPar?

See the description of continuation on the page Library.

How can I change the preconditioner during computations?

The easiest way to achieve this is by using the callbacks provided by newton and continuation. See the documentation about these two methods. See also the example 2d Ginzburg-Landau equation

How can I implement my own bifurcation detection method?

You can use the callback finalise_solution but the best way is probably to use the Iterator Interface to inject your code anywhere in the continuation procedure.

How do I dissociate the computation of eigenvalues from the jacobian that I passed?

Sometimes, for example when implementing boundary conditions, you pass a jacobian J but the eigenvalues, and the bifurcation points are not simply related to J. One way to bypass this issue is to define a new eigensolver <: AbstractEigenSolver and pass it to the NewtonPar field eigsolver. This is done for example in example/SH2d-fronts-cuda.jl.

How can I print the eigenvalues during continuation?

You can print the eigenvalues using the following callback:

finalise_solution = (z, tau, step, contResult; k...) -> begin
		BK.haseigenvalues(contResult) && Base.display(contResult.eig[end].eigenvals)
		return true

How can I reject a Newton Step?

You can reject a newton step by passing to continuation the argument callback_newton

function mycallback((x, f, J, res, iteration, itlinear, options); kwargs...)
	# stop Newton algo if residual too large
	if res > 1e2
		@warn "Reject Newton step!!"
		return false
	return true

How do I stop continuation?

Using the argument finalise_solution in continuation. Simply make this function finalise_solution return false.

How do I compute both sides of a branch?

Instead of using two calls to continuation, you can pass the keyword bothside=true to continuation

How do I compute period orbits for non-autonomous problems

The package does not yet allow to compute periodic orbits solutions of non-autonomous Cauchy problems like

\[\frac{du}{dt} = F(u, par, t).\]

On certain cases, one can still go away with the following trick. Say one is interested (dummy example!) to study

\[\dot u = cos(u) + cos(\omega \cdot t).\]

Then one can use the following autonomous vector field

function vectorField(U, par)
	u, x, y = U
	out = similar(U)
	out[1] = cos(u) + x
	x2 = x^2+y^2
	out[2] = x + par.ω * y - x * x2
	out[3] = y - par.ω * x - y * x2

Arpack is slow in computing eigenvalues

This is probably due to iterative refinement conducted by SuiteSparse as explained in this blog post. You can disable this using

using SuiteSparse
SuiteSparse.UMFPACK.umf_ctrl[8] = 0

Should I use CVODE_BDF?

SciML is now able to match the performance of the Sundials solver CVODE_BDF. Check the news for more information.