🟠 1d Langmuir–Blodgett transfer model
In this tutorial, we try to replicate some of the results of the amazing paper [Köpf]. This example is quite a marvel in the realm of bifurcation analysis, featuring a harp-like bifurcation diagram. The equations of the thin film are as follows:
\[\partial_{t} c=-\partial_{x}^{2}\left[\partial_{x}^{2} c-c^{3}+c-\mu \zeta(x)\right]-V \partial_{x} c\]
with boundary conditions
\[c(0)=c_{0}, \quad \partial_{x x} c(0)=\partial_{x} c(L)=\partial_{x x} c(L)=0\]
and where
\[\zeta(x)=-\frac{1}{2}\left[1+\tanh \left(\frac{x-x_{s}}{l_{s}}\right)\right].\]
As can be seen in the reference above, the bifurcation diagram is significantly more involved as $L$ increases. So we set up for the "simple" case $L=50$.
using Revise
using SparseArrays
using BifurcationKit, LinearAlgebra, Plots, ForwardDiff
const BK = BifurcationKit
# norms
normL2(x; r = sqrt(par.Δx / L)) = norm(x, 2) * r
Let us define the parameters of the model
# domain size
L = 50.0
# number of unknowns
N = 390*3/2 |> Int
Δx = L/(N+1)
X = ((1:N) |> collect) .* Δx
# define the (laplacian of) g function
xs = 10.0; ls = 2.0
Δg = @. tanh((X - xs)/ls) * (1 - tanh((X - xs)/ls)^2)/ls^2
# define the parameters of the model
par = (N = N, Δx = Δx, c0 = -0.9, σ = 1.0, μ = 0.5, ν = 0.08, Δg = Δg)
Encoding the PDE
# function to enforce the boundary condition
function putBC!(c, c0, N)
# we put boundary conditions using ghost points
# this boundary condition u''(0) = 0 = c1 -2c0 + c-1 gives c-1:
c[1] = 2c0-c[3]
# c(0) = c0, we would like to write x[0]
c[2] = c0
# the boundary conditions u'(L) = u''(L) = 0 imply the ghost points values.
# c'(L) = 0 = cN+2 - cN and c''(L) = 0 = cN+2 -2cN+1 + cN
c[N+3] = c[N+2]
c[N+4] = c[N+2]
return c
end
# implementation of the right hand side of the PDE
function Flgvf!(out, x, p, t = 0.)
(;c0, N, Δx, σ, μ, Δg, ν) = p
dx4 = Δx^4
dx2 = Δx^2
# we declare the residual
# we enforce the BC
c = similar(x, length(x) + 4)
c[3:N+2] .= x
putBC!(c, c0, N)
for i=3:N+2
out[i-2] = -(σ * (c[i-2] - 4c[i-1] + 6c[i] - 4c[i+1] + c[i+2]) / dx4 +
(c[i-1] - 2c[i] + c[i+1]) / (dx2) -
(c[i-1]^3 - 2c[i]^3 + c[i+1]^3) / (dx2) -
Δg[i-2] * μ +
ν * (c[i+1] - c[i-1]) / (2Δx)
)
end
return out
end
Flgvf(x, p, t = 0) = Flgvf!(similar(x), x, p, t)
# compute the jacobian of the PDE at position x
@views function JanaSP(x, p)
# 63.446 μs (61 allocations: 137.97 KiB) pour N = 400
# 62.807 μs (44 allocations: 168.58 KiB) pour sparse(Jana(x, p))
(;N, Δx, σ, ν) = p
d0 = @. (-6σ/ Δx^4 + 2/ Δx^2*(1-3x^2))
d0[1] += σ/ Δx^4
d0[end] = -(3σ/ Δx^4 - 1/ Δx^2*(1-3x[N]^2) + ν/ (2Δx))
d1 = @. (4σ/ Δx^4 - 1/ Δx^2*(1-3x[2:N]^2) - ν/ (2Δx))
dm1 = @. (4σ/ Δx^4 - 1/ Δx^2*(1-3x[1:N-1]^2) + ν/ (2Δx))
d1[end] -= σ/ Δx^4
d2 = @. (-σ/ Δx^4) * ones(N-2)
J = spdiagm( 0 => d0,
1 => d1,
-1 => dm1,
2 => d2,
-2 => d2)
return J
end
# we thus define a bifurcation problem
prob = BifurcationProblem(Flgvf, 0X .-0.9, par, (@optic _.ν );
J = JanaSP,
record_from_solution = (x, p; k...) -> normL2(x),
plot_solution = (x, p; kwargs...) -> plot!(X, x, subplot = 3, xlabel = "Nx = $(length(x))", label = ""))
Continuation of stationary states
We call the Krylov-Newton method to find a stationary solution. Note that for this to work, the guess has to satisfy the boundary conditions approximately.
# newton iterations to refine the guess
opt_new = NewtonPar(tol = 1e-9, verbose = true, max_iterations = 10)
out = BK.solve(prob, Newton(), opt_new)
┌─────────────────────────────────────────────────────┐
│ Newton step residual linear iterations │
├─────────────┬──────────────────────┬────────────────┤
│ 0 │ 3.1252e-01 │ 0 │
│ 1 │ 1.0171e-01 │ 1 │
│ 2 │ 9.8336e-03 │ 1 │
│ 3 │ 8.6535e-05 │ 1 │
│ 4 │ 6.3118e-09 │ 1 │
│ 5 │ 1.8452e-10 │ 1 │
└─────────────┴──────────────────────┴────────────────┘
scene = plot(X, out.u)
We then continue the previous guess and find this very nice folded structure with many Hopf bifurcation points.
# careful here, in order to use Arpack.eig, you need rather big space
# or compute ~100 eigenvalues
opts_cont = ContinuationPar(
p_min = -0.01, p_max = 10.1,
dsmin = 1e-5, dsmax = 0.04, ds= -0.001,
a = 0.75, max_steps = 600,
newton_options = NewtonPar(opt_new; verbose = false),
nev = 10, save_eigenvectors = true, tol_stability = 1e-5, detect_bifurcation = 3,
dsmin_bisection = 1e-8, max_bisection_steps = 15, n_inversion = 6, tol_bisection_eigenvalue = 1e-9, save_sol_every_step = 50)
# we opt for a fast Shift-Invert eigen solver
@reset opts_cont.newton_options.eigsolver = EigArpack(0.1, :LM)
br = @time continuation(
re_make(prob, params = (@set par.ν = 0.06), u0 = out.u),
# we form a sparse matrix for the bordered linear problem
# and we adjust θ so that the continuation steps are larger
PALC(θ = 0.4, bls = MatrixBLS()), opts_cont,
plot = true, verbosity = 2,
normC = normL2)
scene = plot(br, title="N=$N")
plot(layout = grid(4, 3))
for (ii, s) in pairs(br.sol)
plot!(X, s.x, xlabel = "ν = $(round(s.p,digits=3))", subplot = ii, label="",tickfont = (7, :black), ylims=(-1,1.5))
end
title!("")
Continuation of Hopf and Fold points
Let us study the continuation of Hopf and Fold points and show that they merge at a Bogdanov-Takens bifurcation point:
# compute branch of Fold points from 7th bifurcation point on br
sn_codim2 = continuation(br, 7, (@optic _.Δx),
ContinuationPar(opts_cont, p_min = -2, p_max = 0.12, ds = -0.01, dsmax = 0.01, tol_stability = 1e-8, max_steps = 325, nev=23) ;
# start the problem with information from eigen elements
start_with_eigen = true,
# detection of codim 2 bifurcations with bisection
detect_codim2_bifurcation = 2,
# we update the Fold problem at every continuation step
update_minaug_every_step = 1,
# compute both sides of the initial condition
bothside = true,
# we invert the Fold linear problem using Min. Aug.
jacobian_ma = :minaug,
)
# compute branch of Hopf points from 5th bifurcation point on br
hp_codim2 = continuation(br, 5, (@optic _.Δx), ContinuationPar(opts_cont, p_max = 0.1, ds = -0.01, dsmax = 0.01, max_steps = 230, tol_stability = 1e-8) ;
# start the problem with information from eigen elements
start_with_eigen = true,
# we update the Hopf problem at every continuation step
update_minaug_every_step = 1,
# detection of codim 2 bifurcations with bisection
detect_codim2_bifurcation = 2,
# we invert the Hopf linear problem using Min. Aug.
jacobian_ma = :minaug,
)
# plot the branches
plot(sn_codim2, vars = (:Δx, :ν), branchlabel = "Fold")
plot!(hp_codim2, vars = (:Δx, :ν), branchlabel = "Hopf", plotcirclesbif=true, legend = :bottomright, color = :green)
Continuation of periodic orbits (FD)
We would like to compute the branches of periodic solutions from the Hopf points. We do this automatic branch switching as follows
# parameters for newton
opt_po = NewtonPar(tol = 1e-10, verbose = true, max_iterations = 50)
# parameters for continuation
opts_po_cont = ContinuationPar(dsmin = 1e-5, dsmax = 0.35, ds= -0.001,
p_max = 1.0, max_steps = 3, detect_bifurcation = 0,
newton_options = NewtonPar(opt_po; max_iterations = 15, tol = 1e-6), plot_every_step = 1)
# spatio-temporal norm
normL2T(x; r = sqrt(par.Δx / L), M = 1) = norm(x, 2) * r * sqrt(1/M)
M = 100 # number of time sections
br_potrap = continuation(
# arguments for branch switching
br, 5,
# arguments for continuation
opts_po_cont, PeriodicOrbitTrapProblem(M = M, jacobian = :FullSparseInplace, update_section_every_step = 1);
# parameter value used for branching
δp = 1e-5,
# use deflated Newton to find non-trivial solutions
usedeflation = true,
# algorithm to solve linear associated with periodic orbit problem
# tangent algorithm along the branch, linear algo specific to PALC
alg = PALC(tangent = Bordered(), bls = BorderingBLS(solver = DefaultLS(), check_precision = false)),
verbosity = 3, plot = true,
record_from_solution = (x, p; k...) -> normL2T(x[1:end-1], M = M),
plot_solution = (x, p; kwargs...) -> begin
heatmap!(reshape(x[1:end-1], N, M)'; ylabel="T=$(round(x[end]))", color=:viridis, kwargs...)
plot!(br, subplot=1, label="")
end,
normC = norminf)
and we obtain the following graph. It is interesting to note that the periodic solutions converge to an homoclinic orbit here with a very large period.
We can do this for the other Hopf points as well. Note that, we have to increase the number of time sections M
to improve the convergence to the homoclinic orbits.
Here are some examples of periodic solutions.
- Köpf
Köpf and Thiele, Emergence of the Bifurcation Structure of a Langmuir–Blodgett Transfer Model., 2014