using CriticalTransitions
using StaticArrays
using Logging

Maier-Stein β = 10: drift is a 2D vector field with three fixed points, attractors at (±1, 0) and a saddle at (0, 0). The instanton path from attractor → attractor crosses the saddle, so it cannot be solved as a single shooting BVP (the arclength reparameterization develops a singularity wherever H_p = 0, which on the H = 0 shell forces (x, p) = (x*, 0) at any drift fixed point). The user must split into two attractor → saddle legs.

function maier_stein(u, p, t)
    x, y = u
    return SA[x - x^3 - 10 * x * y^2, -(1 + x^2) * y]
end

ds = CoupledSDEs(maier_stein, zeros(2); noise_strength = 0.25)
H = FreidlinWentzellHamiltonian(ds)

Freidlin-Wentzell Hamiltonian on 2-dimensional state space (constant Diagonal a) with out-of-place H_x and H_p

Reference: gMAM on the full attractor → attractor path.

Nt = 30
xx = collect(range(-1.0, 1.0; length = Nt))
yy = 0.3 .* (1 .- xx .^ 2)
x_initial_full = Matrix([xx yy]')

res_gmam = minimize_geometric_action(
    H, x_initial_full, GeometricGradient(; stepsize = 1.0);
    maxiters = 500, show_progress = false,
)
println(
    "gMAM action (full path, deterministic return makes second half free):  ",
    res_gmam.action
)

gMAM action (full path, deterministic return makes second half free):  0.3370927697573046

Shooting: left half (-1, 0) → (0, 0).

xs_left = collect(range(-1.0, 0.0; length = Nt))
ys_left = 0.3 .* sin.(π .* (xs_left .+ 1))
x_init_left = Matrix([xs_left ys_left]')

res_left = with_logger(NullLogger()) do
    minimize_geometric_action(
        H, x_init_left,
        MultipleShooting(; nshoots = 8, maxiters = 200, abstol = 1.0e-6);
        show_progress = false,
    )
end
println("Shooting action (-1, 0) → (0, 0):  ", res_left.action)

Shooting action (-1, 0) → (0, 0):  0.33929398443223363

Right half by x → -x symmetry: (+1, 0) → (0, 0) has equal action.

xs_right = collect(range(1.0, 0.0; length = Nt))
ys_right = 0.3 .* sin.(π .* (1 .- xs_right))
x_init_right = Matrix([xs_right ys_right]')

res_right = with_logger(NullLogger()) do
    minimize_geometric_action(
        H, x_init_right,
        MultipleShooting(; nshoots = 8, maxiters = 200, abstol = 1.0e-6);
        show_progress = false,
    )
end
println("Shooting action (+1, 0) → (0, 0):  ", res_right.action)

Shooting action (+1, 0) → (0, 0):  0.33929398443223363

In Freidlin-Wentzell theory the saddle → attractor leg follows the deterministic drift and contributes zero action. The full transition action is therefore the maximum of the two halves (in the symmetric case they are equal). gMAM's full-path action above should match either half.

println(
    "Symmetry check: |left - right| / max = ",
    abs(res_left.action - res_right.action) / max(res_left.action, res_right.action)
)

Symmetry check: |left - right| / max = 0.0

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