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🟑 Extended Lorenz-84 model (codim 2 + BT/ZH aBS)

In this tutorial, we study the extended Lorenz-84 model which is also treated in MatCont [Kuznetsov]. This model is interesting because it features all codim 2 bifurcations of equilibria. It is thus convenient to test our algorithms.

After this tutorial, you will be able to

  • detect codim 1 bifurcation Fold / Hopf / Branch point
  • follow Fold / Hopf points and detect codim 2 bifurcation points
  • branch from the codim 2 points to curves of Fold / Hopf points

The model is as follows

\[\left\{\begin{array}{l} \dot{X}=-Y^{2}-Z^{2}-\alpha X+\alpha F-\gamma U^{2} \\ \dot{Y}=X Y-\beta X Z-Y+G \\ \dot{Z}=\beta X Y+X Z-Z \\ \dot{U}=-\delta U+\gamma U X+T \end{array}\right.\tag{E}\]

We start with some imports:

using Revise, Plots
using BifurcationKit
const BK = BifurcationKit

Problem setting

We can now encode the vector field (E) in a function.

# vector field
function Lor(u, p)
	(;Ξ±,Ξ²,Ξ³,Ξ΄,G,F,T) = p
	X,Y,Z,U = u
	[
		-Y^2 - Z^2 - Ξ±*X + Ξ±*F - Ξ³*U^2,
		X*Y - Ξ²*X*Z - Y + G,
		Ξ²*X*Y + X*Z - Z,
		-Ξ΄*U + Ξ³*U*X + T
	]
end

# parameter values
parlor = (Ξ± = 1//4, Ξ² = 1, G = .25, Ξ΄ = 1.04, Ξ³ = 0.987, F = 1.7620532879639, T = .0001265)

# initial condition
z0 = [2.9787004394953343, -0.03868302503393752,  0.058232737694740085, -0.02105288273117459]

# bifurcation problem
recordFromSolutionLor(x, p) = (X = x[1], Y = x[2], Z = x[3], U = x[4])
prob = BifurcationProblem(Lor, z0, (parlor..., T=0.04, F=3.), (@lens _.F);
    record_from_solution = recordFromSolutionLor)

Continuation and codim 1 bifurcations

Once the problem is set up, we can continue the state w.r.t. $F$ and detect codim 1 bifurcations. This is achieved as follows:

# continuation options
opts_br = ContinuationPar(p_min = -1.5, p_max = 3.0, ds = 0.002, dsmax = 0.15,
	# Optional: bisection options for locating bifurcations
	n_inversion = 6,
	# number of eigenvalues
	nev = 4)

# compute the branch of solutions
br = continuation(prob, PALC(), opts_br;
	normC = norminf,
	bothside = true)

scene = plot(br, plotfold = false, markersize = 4, legend = :topleft)
Example block output

With detailed information:

br
 β”Œβ”€ Curve type: EquilibriumCont
 β”œβ”€ Number of points: 33
 β”œβ”€ Type of vectors: Vector{Float64}
 β”œβ”€ Parameter F starts at 3.0, ends at 3.0
 β”œβ”€ Algo: PALC
 └─ Special points:

- #  1, endpoint at F β‰ˆ +3.00000000,                                                                     step =   0
- #  2,     hopf at F β‰ˆ +2.85996783 ∈ (+2.85986480, +2.85996783), |Ξ΄p|=1e-04, [converged], Ξ΄ = ( 2,  2), step =   1
- #  3,     hopf at F β‰ˆ +2.46723305 ∈ (+2.46720734, +2.46723305), |Ξ΄p|=3e-05, [converged], Ξ΄ = (-2, -2), step =   3
- #  4,     hopf at F β‰ˆ +1.61975642 ∈ (+1.61959602, +1.61975642), |Ξ΄p|=2e-04, [converged], Ξ΄ = ( 2,  2), step =   9
- #  5,       bp at F β‰ˆ +1.54664839 ∈ (+1.54664837, +1.54664839), |Ξ΄p|=1e-08, [converged], Ξ΄ = (-1,  0), step =  11
- #  6, endpoint at F β‰ˆ +3.00000000,                                                                     step =  32

Continuation of Fold points

We follow the Fold points in the parameter plane $(T,F)$. We tell the solver to consider br.specialpoint[5] and continue it.

# function to record the current state
sn_codim2 = continuation(br, 5, (@lens _.T),
	ContinuationPar(opts_br, p_max = 3.2, p_min = -0.1,
		dsmin=1e-5, ds = -0.001, dsmax = 0.005) ;
	normC = norminf,
	# detection of codim 2 bifurcations with bisection
	detect_codim2_bifurcation = 2,
	start_with_eigen = false,
	# we save the different components for plotting
	record_from_solution = recordFromSolutionLor,
	)

scene = plot(sn_codim2, vars=(:X, :U), branchlabel = "Folds", ylims=(-0.5, 0.5))
Example block output

with detailed information

sn_codim2
 β”Œβ”€ Curve type: FoldCont
 β”œβ”€ Number of points: 82
 β”œβ”€ Type of vectors: Vector{Float64}
 β”œβ”€ Parameters (:F, :T)
 β”œβ”€ Parameter T starts at 0.04, ends at -0.1
 β”œβ”€ Algo: PALC
 └─ Special points:

- #  1,       bt at T β‰ˆ +0.02094014 ∈ (+0.02094014, +0.02094020), |Ξ΄p|=6e-08, [converged], Ξ΄ = ( 0,  0), step =  12
- #  2,       zh at T β‰ˆ +0.00012644 ∈ (+0.00012644, +0.00012666), |Ξ΄p|=2e-07, [converged], Ξ΄ = ( 0,  0), step =  29
- #  3,       zh at T β‰ˆ -0.00012655 ∈ (-0.00012655, -0.00012644), |Ξ΄p|=1e-07, [converged], Ξ΄ = ( 0,  0), step =  32
- #  4,       bt at T β‰ˆ -0.02094041 ∈ (-0.02094041, -0.02093949), |Ξ΄p|=9e-07, [converged], Ξ΄ = ( 0,  0), step =  49
- #  5, endpoint at T β‰ˆ -0.10000000,                                                                     step =  81

For example, we can compute the following normal form

get_normal_form(sn_codim2, 1; nev = 4)
Bogdanov-Takens bifurcation point at (:F, :T) β‰ˆ (1.4467165285461494, 0.020940139624099376).
Normal form (B, Ξ²1 + Ξ²2β‹…B + bβ‹…Aβ‹…B + aβ‹…AΒ²)
Normal form coefficients:
 a = 0.2144232743197446
 b = 0.6065142321261716

You can call various predictors:
 - predictor(::BogdanovTakens, ::Val{:HopfCurve}, ds)
 - predictor(::BogdanovTakens, ::Val{:FoldCurve}, ds)
 - predictor(::BogdanovTakens, ::Val{:HomoclinicCurve}, ds)

Continuation of Hopf points

We follow the Hopf points in the parameter plane $(T,F)$. We tell the solver to consider br.specialpoint[3] and continue it.

hp_codim2_1 = continuation(br, 3, (@lens _.T),
	ContinuationPar(opts_br, ds = -0.001, dsmax = 0.02, dsmin = 1e-4) ;
	normC = norminf,
	# detection of codim 2 bifurcations with bisection
	detect_codim2_bifurcation = 2,
	# we save the different components for plotting
	record_from_solution = recordFromSolutionLor,
	# compute both sides of the initial condition
	bothside = true,
	)

plot(sn_codim2, vars=(:X, :U), branchlabel = "Folds")
plot!(hp_codim2_1, vars=(:X, :U), branchlabel = "Hopfs")
ylims!(-0.7,0.7);xlims!(1,1.3)
Example block output
hp_codim2_1
 β”Œβ”€ Curve type: HopfCont
 β”œβ”€ Number of points: 429
 β”œβ”€ Type of vectors: Vector{Float64}
 β”œβ”€ Parameters (:F, :T)
 β”œβ”€ Parameter T starts at 0.020940169755227573, ends at -0.1535694927465657
 β”œβ”€ Algo: PALC
 └─ Special points:

- #  1, endpoint at T β‰ˆ +0.02094017,                                                                     step =   0
- #  2,       bt at T β‰ˆ +0.02094017 ∈ (+0.02094017, +0.02094017), |Ξ΄p|=6e-11, [converged], Ξ΄ = ( 0,  0), step =   0
- #  3,       gh at T β‰ˆ +0.05019751 ∈ (+0.05019655, +0.05019751), |Ξ΄p|=1e-06, [converged], Ξ΄ = ( 0,  0), step =  19
- #  4,       hh at T β‰ˆ +0.02627340 ∈ (+0.02627340, +0.02627528), |Ξ΄p|=2e-06, [converged], Ξ΄ = (-2, -2), step =  35
- #  5, endpoint at T β‰ˆ -0.15372759,                                                                     step = 429

For example, we can compute the following normal form

get_normal_form(hp_codim2_1, 3; nev = 4)
Bautin bifurcation point at (:F, :T) β‰ˆ (2.3763590366726284, 0.05019751302158527).
Ο‰ = 0.6903670769045964
Second lyapunov coefficient lβ‚‚ = 0.1557753180139064
Normal form: iβ‹…Ο‰β‹…z + lβ‚‚β‹…zβ‹…|z|⁴

Continuation of Hopf points from the Bogdanov-Takens point

When we computed the curve of Fold points, we detected a Bogdanov-Takens bifurcation. We can branch from it to get the curve of Hopf points. This is done as follows:

hp_from_bt = continuation(sn_codim2, 4,
	ContinuationPar(opts_br, ds = -0.001, dsmax = 0.02, dsmin = 1e-4) ;
	normC = norminf,
	# detection of codim 2 bifurcations with bisection
	detect_codim2_bifurcation = 2,
	# we save the different components for plotting
	record_from_solution = recordFromSolutionLor,
	)

plot(sn_codim2, vars=(:X, :U), branchlabel = "SN")
plot!(hp_codim2_1, vars=(:X, :U), branchlabel = "Hopf1")
plot!(hp_from_bt, vars=(:X, :U), branchlabel = "Hopf2")
ylims!(-0.7,0.75); xlims!(0.95,1.3)
Example block output

with detailed information

hp_from_bt
 β”Œβ”€ Curve type: HopfCont from BogdanovTakens bifurcation point.
 β”œβ”€ Number of points: 401
 β”œβ”€ Type of vectors: Vector{Float64}
 β”œβ”€ Parameters (:F, :T)
 β”œβ”€ Parameter T starts at -0.026824826717918287, ends at 0.15063717399994617
 β”œβ”€ Algo: PALC
 └─ Special points:

- #  1,       gh at T β‰ˆ -0.05018361 ∈ (-0.05022014, -0.05018361), |Ξ΄p|=4e-05, [converged], Ξ΄ = ( 0,  0), step =  23
- #  2,       hh at T β‰ˆ -0.02626030 ∈ (-0.02631231, -0.02626030), |Ξ΄p|=5e-05, [converged], Ξ΄ = (-2, -2), step =  26
- #  3, endpoint at T β‰ˆ +0.15079813,                                                                     step = 401

Continuation of Hopf points from the Zero-Hopf point

When we computed the curve of Fold points, we detected a Zero-Hopf bifurcation. We can branch from it to get the curve of Hopf points. This is done as follows:

hp_from_zh = continuation(sn_codim2, 2,
	ContinuationPar(opts_br, ds = 0.001, dsmax = 0.02) ;
	normC = norminf,
	detect_codim2_bifurcation = 2,
	record_from_solution = recordFromSolutionLor,
	)

plot(hp_codim2_1, vars=(:X, :U), branchlabel = "Hopf")
plot!(hp_from_bt, vars=(:X, :U),  branchlabel = "Hopf2")
plot!( hp_from_zh, vars=(:X, :U), branchlabel = "Hopf", legend = :topleft)
plot!(sn_codim2,vars=(:X, :U),)
ylims!(-0.7,0.75); xlims!(0.95,1.3)
Example block output

with detailed information

hp_from_zh
 β”Œβ”€ Curve type: HopfCont from BifurcationKit.ZeroHopf bifurcation point.
 β”œβ”€ Number of points: 401
 β”œβ”€ Type of vectors: Vector{Float64}
 β”œβ”€ Parameters (:F, :T)
 β”œβ”€ Parameter T starts at 0.0001264399489437422, ends at 0.666963445643025
 β”œβ”€ Algo: PALC
 └─ Special points:

- #  1,       gh at T β‰ˆ +0.00012660 ∈ (+0.00012654, +0.00012660), |Ξ΄p|=6e-08, [converged], Ξ΄ = ( 0,  0), step =   1
- #  2,       hh at T β‰ˆ +0.02627444 ∈ (+0.02627324, +0.02627444), |Ξ΄p|=1e-06, [converged], Ξ΄ = ( 2,  2), step =  27
- #  3, endpoint at T β‰ˆ +0.66778051,                                                                     step = 401

References

  • Kuznetsov

    Kuznetsov, Yu A., H. G. E. Meijer, W. Govaerts, and B. Sautois. β€œSwitching to Nonhyperbolic Cycles from Codim 2 Bifurcations of Equilibria in ODEs.” Physica D: Nonlinear Phenomena 237, no. 23 (December 2008): 3061–68.