Question: detect fold and branch point separately

[rdprins]

When generating a bifurcation diagram (1 param), both folds and branch points are give the type 'bp'. Is there a way to differentiate between them?

For more context, this is my finalise_solution function, which saves the results. I would like the variable TY to be either 'fold' or 'branch point' when applicable.

function append_file(filename, s)
    open(filename, "a") do file
        println(file,s)
    end
end;

function finalise_solution(z, tau, step, br, filename)
    if (length(br.specialpoint)>0) && (br.specialpoint[end].step == step)
        TY = br.specialpoint[end].type
        special_type=true
    else
        TY = "RG"
        special_type=false
    end

    if special_type || (step % 10 == 0)
        stability = br.stable[end]
        append_file(filename, join([step, z.p, TY, stability, z.u...], ";"))
    end

    return true
end;

[rveltz]

Hi,

Sorry, this is tricky at the moment as I dont use events to locate bifurcation (as MatCont do). If you can call get_normal_form(br, ind_fold), that would help you for example doing get_normal_form(br, ind_fold) isa BK.Fold

[rdprins]

Thank you for your quick reply!

Unfortunately it isn't possible to call get_normal_form(br, length(br.specialpoint), nev=nev) within finalise_solution, because an error occurs in NormalForms.jl Specifically, at line 80: λ = real(br.eig[bifpt.idx].eigenvals[bifpt.ind_ev]) indexing goes out of bounds.

BoundsError: attempt to access 45-element Vector{@NamedTuple{eigenvals::Vector{ComplexF64}, eigenvecs::Matrix{ComplexF64}, converged::Bool, step::Int64}} at index [46]

Would you know how to resolve this?

[rveltz]

Here is another way based on events. The idea is to locate Folds (using FoldDetectEvent) and bifurcations (using BifDetectEvent)

You will see points like foldbp because the event is a Fold and a bifurcation point at the same time. This could be resolved at a later stage by changing :foldbp to fold

I hope this is enough for you, sorry for the inconvenience. Because I want to be able to address bifurcations of kernel dimension >1 like in PDE, I do not use the event machinery of MatCont/Auto but just check for change in the number of unstable eigenvalues.

opts_br = ContinuationPar(p_min = -10.0, p_max = -0., dsmax = 0.1, n_inversion = 6, nev = 3, detect_event = 2, detect_bifurcation = 0, max_steps = 130, detect_fold = false)
br = continuation(prob, PALC(tangent = Bordered()), opts_br; 
        normC = norminf, 
        # event = BK.FoldDetectEvent,
        # event = BK.BifDetectEvent,
        event = BK.PairOfEvents(BK.FoldDetectEvent, BK.BifDetectEvent),
        bothside = false)

[rveltz]

Unfortunately it isn't possible to call get_normal_form(br, length(br.specialpoint), nev=nev) within finalise_solution, because an error occurs in NormalForms.jl Specifically, at line 80: λ = real(br.eig[bifpt.idx].eigenvals[bifpt.ind_ev]) indexing goes out of bounds.

Just to be sure. Can you call get_normal_form(br, ind) once the branch has been computed without your finalizer?

[rdprins]

Here is another way based on events. The idea is to locate Folds (using FoldDetectEvent) and bifurcations (using BifDetectEvent)

You will see points like foldbp because the event is a Fold and a bifurcation point at the same time. This could be resolved at a later stage by changing :foldbp to fold

I hope this is enough for you, sorry for the inconvenience. Because I want to be able to address bifurcations of kernel dimension >1 like in PDE, I do not use the event machinery of MatCont/Auto but just check for change in the number of unstable eigenvalues.

opts_br = ContinuationPar(p_min = -10.0, p_max = -0., dsmax = 0.1, n_inversion = 6, nev = 3, detect_event = 2, detect_bifurcation = 0, max_steps = 130, detect_fold = false)
br = continuation(prob, PALC(tangent = Bordered()), opts_br; 
        normC = norminf, 
        # event = BK.FoldDetectEvent,
        # event = BK.BifDetectEvent,
        event = BK.PairOfEvents(BK.FoldDetectEvent, BK.BifDetectEvent),
        bothside = false)

This works, thanks so much!

I have two additional questions:

• Do I understand correctly that BK.Bordered will at least be equally correct as BK.Secant, but might take some additional time to compute?
• Will bifurcationdiagram() work exactly the same after setting detect_bifurcation = 0? (Which is needed because detect_bifurcation = 3 and detect_event = 2 aren't compatible). E.g., do we still use the bisection algorithm to calculate branch points (using e.g. the set values for n_inversion and max_bisection_steps)?

In other words, can I always expect the same behaviour as when using bifurcationdiagram(PALC(tangent = Secant()), detect_bifurcation = 3, detect_event=0, detect_fold=true)?

[rveltz]

Do I understand correctly that BK.Bordered will at least be equally correct as BK.Secant, but might take some additional time to compute?

Yes it is a more precise tangent predictor, takes more time to compute than Secant. FoldDetectEvent does not work well with Secant predictor

[rveltz]

Will bifurcationdiagram() work exactly the same after setting detect_bifurcation = 0? (Which is needed because detect_bifurcation = 3 and detect_event = 2 aren't compatible). E.g., do we still use the bisection algorithm to calculate branch points (using e.g. the set values for n_inversion and max_bisection_steps)?

Yes, you can pass events to bifurcationdiagram the same way you do for continuation

[rveltz]

Ill close this issue. Re-open if you have any other problem.