Only trivial solution in BifurcationKit

Specific Domains Numerics
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1

I try to investigate pitchfork bifurcation, using the BifurcationKit. My problem looks as follow. I start from some definitions:

ND = 500
dx = 1.0/ND
x = collect(1:ND)*dx.-dx/2

J = +1.0
beta0 = 0.1
u0 = 0.01*ones(Float64,ND)
W = ones(Float64,ND,ND);

function rhs(u, beta)
    global J,dx,W
    return u - tanh.(J*beta*dx*W*u)
end

Then, I introduce bifurcation problem:

prob = BifurcationKit.BifurcationProblem(rhs, u0, beta0, 1)

and use

opt_newton = BifurcationKit.NewtonPar(tol=1e-9)
opts = BifurcationKit.ContinuationPar(p_min=0.0, p_max=5.0, max_steps=500, dsmax=2.e-2, dsmin=1.e-4, ds=1.e-4, nev=ND, newton_options=opt_newton);

I run continuation method as follows:

br = BifurcationKit.continuation(prob, BifurcationKit.PALC(),opts, bothside = true, normC=norm, verbosity=0)

I see the correct bifurcation point at p\approx 1 and extract it:

p_test = br.specialpoint[2].param

Then, I try to switch branch:

test = BifurcationKit.continuation(br, 2, opts; δp = -0.1)

My final goal is to obtain 2 vectors p_values and solution_u. I have tried to extract this solution as br.sol or test.sol. But everywhere I see only the (seems that) trivial solution. For instance:

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I have tried to manipulate with record_from_solution, but it does not give the desired result. I have tried to add:

record_from_solution = (u,beta; k...) -> (n2=norm(u),solu=u)

I have read the documentation, but I still do not understand properly what did I wrong.

2

It seems too far from the bifurcation point. This works:

 test = BifurcationKit.continuation(br, 2, opts; δp = -0.01)

I am surprised it does not converged though.

3

If you use debug mode ENV["JULIA_DEBUG"] = BifurcationKit, your code gives:

┌ Debug: The zero eigenvalue is not that small λ = -0.0007633203318768711
│ This can alter the computation of the normal form.
│ You can either refine the point using Newton or use a more precise bisection by increasing `n_inversion`
└ @ BifurcationKit ~/work/prog_gd/julia/dev/dev1/bkorg/BifurcationKitGITHUB/src/NormalForms.jl:88
┌ Debug: The guess for the amplitude of the first periodic orbit on the bifurcated branch obtained by the predictor is not small: 3.8685530832137087. This may lead to convergence failure of the first newton step or select a branch far from the Hopf point.
│ You can either decrease `ds` or `δp` (which is  how far from the bifurcation point you want the branch of periodic orbits to start). Alternatively, you can specify a multiplicative factor `ampfactor` to be applied to the predictor amplitude.
└ @ BifurcationKit ~/work/prog_gd/julia/dev/dev1/bkorg/BifurcationKitGITHUB/src/bifdiagram/BranchSwitching.jl:152
┌ Debug: Update tangent
└ @ BifurcationKit ~/work/prog_gd/julia/dev/dev1/bkorg/BifurcationKitGITHUB/src/continuation/Palc.jl:125
┌ Debug: Saving data on the branch
│   state.step = 1
└ @ BifurcationKit ~/work/p
...

You have the indication You can either decrease dsorδp`` albeit the message is wrong (it is for Hopf bifurcation and I will correct this).