# ๐  Steinmetz-Larter model

This example is found in the MatCont ecosystem.

The Steinmetz-Larter model is studied in the MatCont ecosystem because it is a simple example where a Chenciner bifurcation occurs.

$$$\tag{E}\left\{\begin{array}{l} \dot{A}=-k_1 A B X-k_3 A B Y+k_7-k_{-7} A, \\ \dot{B}=-k_1 A B X-k_3 A B Y+k_8, \\ \dot{X}=k_1 A B X-2 k_2 X^2+2 k_3 A B Y-k_4 X+k_6, \\ \dot{Y}=-k_3 A B Y+2 k_2 X^2-k_5 Y, \end{array}\right.$$$

This tutorial is also useful in that we show how to start periodic orbits continuation from solutions obtained from solving ODEs. Being a relatively advanced tutorial, we do not give too many details.

We start by coding the bifurcation problem.

using Revise
using ForwardDiff, Parameters, Plots
using BifurcationKit
const BK = BifurcationKit

function SL!(du, u, p, t = 0)
@unpack k1, k2, k3, k4, k5, k6, k7, kโโ, k8 = p
A,B,X,Y = u

du[1] = -k1*A*B*X - k3*A*B*Y + k7 - kโโ*A
du[2] = -k1*A*B*X - k3*A*B*Y + k8
du[3] =  k1*A*B*X - 2*k2*X^2 + 2*k3*A*B*Y - k4*X+k6
du[4] = -k3*A*B*Y + 2*k2*X^2 - k5*Y
du
end

z0 = rand(4)
par_sl = (k1=0.1631021, k2=1250., k3=0.046875, k4=20., k5=1.104, k6=0.001, kโโ=0.1175, k7=1.5, k8=0.75)
prob = BifurcationProblem(SL!, z0, par_sl, (@lens _.k8);)

# record variables for plotting
function recordFromSolution(x, p)
xtt = BK.get_periodic_orbit(p.prob, x, p.p)
return (max = maximum(xtt[1,:]),
min = minimum(xtt[1,:]),
period = getperiod(p.prob, x, p.p))
end

# plotting function
function plotSolution(x, p; k...)
xtt = BK.get_periodic_orbit(p.prob, x, p.p)
plot!(xtt.t, xtt[:,:]'; label = "", k...)
end

# group parameters
argspo = (record_from_solution = recordFromSolution,
plot_solution = plotSolution)

We obtain some trajectories as seeds for computing periodic orbits.

using DifferentialEquations
alg_ode = Rodas5()
prob_de = ODEProblem(SL!, z0, (0,136.), par_sl)
sol = solve(prob_de, alg_ode)
prob_de = ODEProblem(SL!, sol.u[end], (0,30.), sol.prob.p, reltol = 1e-11, abstol = 1e-13)
sol = solve(prob_de, alg_ode)
plot(sol)

## Computation with Shooting

We generate a shooting problem from the computed trajectories and continue the periodic orbits as function of $k_8$

probsh, cish = generate_ci_problem( ShootingProblem(M=4), prob, prob_de, sol, 16.; reltol = 1e-10, abstol = 1e-12, parallel = true)

opts_po_cont = ContinuationPar(p_min = 0., p_max = 20.0, ds = 0.002, dsmax = 0.05, n_inversion = 8, detect_bifurcation = 3, max_bisection_steps = 25, nev = 4, max_steps = 60, save_eigenvectors = true, tol_stability = 1e-3)
@set! opts_po_cont.newton_options.verbose = false
@set! opts_po_cont.newton_options.max_iterations = 10
br_sh = continuation(deepcopy(probsh), cish, PALC(tangent = Bordered()), opts_po_cont;
# verbosity = 3, plot = true,
callback_newton = BK.cbMaxNorm(10),
argspo...)
scene = plot(br_sh)

### Curve of Fold points of periodic orbits

opts_posh_fold = ContinuationPar(br_sh.contparams, detect_bifurcation = 2, max_steps = 35, p_max = 1.9, plot_every_step = 10, dsmax = 4e-2, ds = 1e-2)
@set! opts_posh_fold.newton_options.tol = 1e-12
# @set! opts_posh_fold.newton_options.verbose = true
fold_po_sh = @time continuation(deepcopy(br_sh), 2, (@lens _.k7), opts_posh_fold;
# verbosity = 2, plot = true,
detect_codim2_bifurcation = 0,
update_minaug_every_step = 1,
start_with_eigen = true,
usehessian = false,
jacobian_ma = :minaug,
normC = norminf,
callback_newton = BK.cbMaxNorm(1e1),
# bdlinsolver = BorderingBLS(solver = DefaultLS(), check_precision = false),
)
plot(fold_po_sh)

### Curve of NS points of periodic orbits

opts_posh_ns = ContinuationPar(br_sh.contparams, detect_bifurcation = 0, max_steps = 35, p_max = 1.9, plot_every_step = 10, dsmax = 4e-2, ds = 1e-2)
@set! opts_posh_ns.newton_options.tol = 1e-12
# @set! opts_posh_ns.newton_options.verbose = true
ns_po_sh = continuation(deepcopy(br_sh), 1, (@lens _.k7), opts_posh_ns;
# verbosity = 2, plot = true,
detect_codim2_bifurcation = 2,
update_minaug_every_step = 1,
start_with_eigen = false,
jacobian_ma = :minaug,
normC = norminf,
callback_newton = BK.cbMaxNorm(1e1),
)
plot(ns_po_sh, fold_po_sh, branchlabel = ["NS","Fold"])

## Computation with collocation

probcoll, cicoll = generate_ci_problem( PeriodicOrbitOCollProblem(50, 4), prob, sol, 16.)

opts_po_cont = ContinuationPar(p_min = 0., p_max = 2.0,
ds = 0.002, dsmax = 0.05,
# n_inversion = 6,
nev = 4,
max_steps = 50,
tol_stability = 1e-5)
br_coll = continuation(probcoll, cicoll, PALC(tangent = Bordered()), opts_po_cont;
# verbosity = 3, plot = true,
callback_newton = BK.cbMaxNorm(10),
argspo...)
 โโ Curve type: PeriodicOrbitCont
โโ Number of points: 51
โโ Type of vectors: Vector{Float64}
โโ Parameter k8 starts at 0.75, ends at 0.807573529701937
โโ Algo: PALC
โโ Special points:

- #  1,       ns at k8 โ +0.82524727 โ (+0.82478589, +0.82524727), |ฮดp|=5e-04, [converged], ฮด = ( 2,  2), step =  18
- #  2,       bp at k8 โ +0.84391358 โ (+0.84391329, +0.84391358), |ฮดp|=3e-07, [converged], ฮด = (-1,  0), step =  29
- #  3, endpoint at k8 โ +0.80661893,                                                                     step =  51


### Curve of Fold points of periodic orbits

opts_pocl_fold = ContinuationPar(br_coll.contparams, detect_bifurcation = 1, plot_every_step = 10, dsmax = 4e-2)
fold_po_cl = @time continuation(br_coll, 2, (@lens _.k7), opts_pocl_fold;
# verbosity = 3, plot = true,
detect_codim2_bifurcation = 2,
update_minaug_every_step = 1,
start_with_eigen = false,
usehessian = true,
jacobian_ma = :minaug,
normC = norminf,
callback_newton = BK.cbMaxNorm(1e1),
)
 โโ Curve type: FoldPeriodicOrbitCont
โโ Number of points: 51
โโ Type of vectors: BorderedArray{Vector{Float64}, Float64}
โโ Parameters (:k8, :k7)
โโ Parameter k7 starts at 1.5, ends at 1.4576771355437268
โโ Algo: PALC
โโ Special points:

- #  1,     cusp at k7 โ +1.75867266 โ (+1.71740448, +1.75867266), |ฮดp|=4e-02, [    guess], ฮด = ( 0,  0), step =  24
- #  2,     cusp at k7 โ +1.42293437 โ (+1.41990874, +1.42293437), |ฮดp|=3e-03, [converged], ฮด = ( 0,  0), step =  47
- #  3, endpoint at k7 โ +1.46915544,                                                                     step =  51


### Curve of NS points of periodic orbits

opts_pocl_ns = ContinuationPar(br_coll.contparams, detect_bifurcation = 1, plot_every_step = 10, dsmax = 4e-2)
ns_po_cl = continuation(br_coll, 1, (@lens _.k7), opts_pocl_ns;
# verbosity = 3, plot = true,
detect_codim2_bifurcation = 2,
update_minaug_every_step = 1,
start_with_eigen = false,
jacobian_ma = :minaug,
normC = norminf,
callback_newton = BK.cbMaxNorm(1e1),
)
 โโ Curve type: NSPeriodicOrbitCont
โโ Number of points: 47
โโ Type of vectors: BorderedArray{Vector{Float64}, Vector{Float64}}
โโ Parameters (:k8, :k7)
โโ Parameter k7 starts at 1.5, ends at 1.8572289468240268
โโ Algo: PALC
โโ Special points:

- #  1,       ch at k7 โ +1.85584414 โ (+1.85517813, +1.85584414), |ฮดp|=7e-04, [converged], ฮด = ( 0,  0), step =  40
- #  2,       R1 at k7 โ +1.85723125 โ (+1.85723125, +1.85750738), |ฮดp|=3e-04, [    guess], ฮด = ( 0,  0), step =  44
- #  3,       R1 at k7 โ +1.85722895 โ (+1.85722895, +1.85736188), |ฮดp|=1e-04, [    guess], ฮด = ( 0,  0), step =  46
- #  4, endpoint at k7 โ +1.85722895,                                                                     step =  46

plot(ns_po_cl, fold_po_cl, branchlabel = ["NS","Fold"])