docs: add sgMAM example

docs/pages.jl

@@ -4,7 +4,8 @@ pages = [
     "Quickstart" => "quickstart.md",
     "Tutorial" => "examples/tutorial.md",
     "Examples" => Any[
-        "Instanton for the Maier-Stein system" => "examples/gMAM_Maierstein.md"
+        "Anlyses for the Maier-Stein system" => "examples/gMAM_Maierstein.md",
+        "sgMAM for the Kerr Parametric Oscillator" => "examples/sgMAM_KPO.md",
     ],
     "Manual" => Any[
         "Define a CoupledSDEs system" => "man/CoupledSDEs.md",

docs/src/examples/sgMAM_KPO.md

@@ -0,0 +1,93 @@
+We demonstrate the simple geometric minimum action method (sgMAM) on the Kerr parametric oscillator (KPO). The method computes the optimal path between two attractors in the phase space that minimizes the action of the system. It is a simplification of the geometric minimum action method (gMAM) by avoiding the computation of the second order derivatives of the extended Hamiltonian of the optimisation problem.
+
+```@example GMAM
+using CriticalTransitions, CairoMakie
+```
+
+The KPO equation is a nonlinear ordinary differential equation that describes the response of the nonlinear parametrically driven resonator at its dominant resonant condition. The equation of motion are of the form:
+```math
+\dot{\mathbf{x}} = \mathbf{f}(\mathbf{x}) + \sigma\mathbf{ξ}(t)
+```
+where `f` is an autonemous drift function and and we have brownian noise `ξ` with intensity `σ`.
+
+Here we define the define the drift of each seperable variable `u` and `v`. In addition, we hard-code the Jacobian of the drift function. 
+```@example GMAM
+const λ = 3 / 1.21 * 2 / 295
+const ω0 = 1.000
+const ω = 1.000
+const γ = 1 / 295
+const η = 0
+const α = -1
+
+function fu(u, v)
+    return (-4 * γ * ω * u - 2 * λ * v - 4 * (ω0 - ω^2) * v - 3 * α * v * (u^2 + v^2)) /
+           (8 * ω)
+end
+function fv(u, v)
+    return (-4 * γ * ω * v - 2 * λ * u + 4 * (ω0 - ω^2) * u + 3 * α * u * (u^2 + v^2)) /
+           (8 * ω)
+end
+stream(u,v) =  Point2f(fu(u, v), fv(u, v))
+dfvdv(u, v) = (-4 * γ * ω + 6 * α * u * v) / (8 * ω)
+dfudu(u, v) = (-4 * γ * ω - 6 * α * u * v) / (8 * ω)
+dfvdu(u, v) = (-2 * λ + 4 * (ω0 - ω^2) + 9 * α * u^2 + 3 * α * v^2) / (8 * ω)
+dfudv(u, v) = (-2 * λ - 4 * (ω0 - ω^2) - 3 * α * u^2 - 9 * α * v^2) / (8 * ω)
+```
+
+The optimisation is performed in a doubled phase space, i.e., every variable of the SDE system is considered as a generelised coordinate $\mathbf{x}$ and gets a corresponding generalised momentum $\mathbf{p}$. The makes it that also systems with dissipative flow can be solved. As such, we extend the phase space by defining the hamiltionian
+```math
+H = \sum_i \frac{p_i^2}{2} + f_i(\mathbf{x})p_i
+```
+Hence, to use the sgMAM method, we need to define the derivatives of the Hamiltonian with respect to the phase space coordinates and the generalised momentum: 
+```@example GMAM
+function H_x(x, p) # ℜ² → ℜ²
+    u, v = eachrow(x)
+    pu, pv = eachrow(p)
+
+    H_u = @. pu * dfudu(u, v) + pv * dfvdu(u, v)
+    H_v = @. pu * dfudv(u, v) + pv * dfvdv(u, v)
+    return Matrix([H_u H_v]')
+end
+function H_p(x, p) # ℜ² → ℜ²
+    u, v = eachrow(x)
+    pu, pv = eachrow(p)
+
+    H_pu = @. pu + fu(u, v)
+    H_pv = @. pv + fv(u, v)
+    return Matrix([H_pu H_pv]')
+end
+
+sys = SgmamSystem(H_x, H_p)
+```
+We saved this function in the `SgmamSystem` struct. We want to find the optimal path between two attractors in the phase space. We define the initial trajectory as `wiggle` between the two attractors.
+```@example GMAM
+# setup
+Nt = 500  # number of discrete time steps
+s = collect(range(0; stop=1, length=Nt))
+
+xa = [-0.0208, 0.0991]
+xb = -xa
+xsaddle = [0.0, 0.0]
+
+# Initial trajectory
+xx = @. (xb[1] - xa[1]) * s + xa[1] + 4 * s * (1 - s) * xsaddle[1]
+yy = @. (xb[2] - xa[2]) * s + xa[2] + 4 * s * (1 - s) * xsaddle[2] + 0.01 * sin(2π * s)
+x_initial = Matrix([xx yy]')
+```
+The optimisation is the performed by the `sgmam` function:
+```@example GMAM
+x_min, S_min, lambda, p, xdot = sgmam(
+    sys, x_initial; iterations=1_000, ϵ=10e2, show_progress=false
+)
+```
+The function returns the optimal path `x_min`, the minmal action `S_min`, the Lagrange multipliers `lambda` associated with the optimal path, the optimal generalised momentum `p`, and the time derivative of the optimal path `xdot`. We can plot the initial trajectory and the optimal path:
+```@example GMAM
+fig, ax, _ = lines(x_initial[1, :], x_initial[2, :]; label="init", linewidth=3, color=:black)
+lines!(x_min[1, :], x_min[2, :]; label="MLP", linewidth=3, color=:red)
+streamplot!(ax, stream, (-0.08, 0.08), (-0.15, 0.15);
+    gridsize=(20, 20), arrow_size=10, stepsize=0.001,
+    colormap=[:gray, :gray]
+)
+axislegend(ax)
+fig
+```

src/largedeviations/sgMAM.jl

@@ -27,16 +27,20 @@ end
 $(TYPEDSIGNATURES)
 
 Performs the simplified geometric Minimal Action Method (sgMAM) on the given system `sys`.
+Our implementation is only valid for additive noise.
 
 This method computes the optimal path in the phase space of a Hamiltonian system that
-minimizes the action. The Hamiltonian functions `H_x` and `H_p` define the system's dynamics
-in a doubled phase. The initial state `x_initial` is evolved iteratively using constrained
-gradient descent with step size parameter `ϵ` over a specified number of iterations.
-The method can display a progress meter and will stop early if the relative tolerance
-`reltol` is achieved.
+minimizes the Freidlin–Wentzell action. The Hamiltonian functions `H_x` and `H_p` define
+the system's dynamics in a doubled phase. The initial state `x_initial` is evolved
+iteratively using constrained gradient descent with step size parameter `ϵ` over a specified
+number of iterations. The method can display a progress meter and will stop early if the
+relative tolerance `reltol` is achieved.
 
 The function returns a tuple containing the final state, the action value,
-the Lagrange multipliers, the momentum, and the state derivatives.
+the Lagrange multipliers, the momentum, and the state derivatives. The implementation is
+based on the work of [Grafke et al. (2019)](
+https://homepages.warwick.ac.uk/staff/T.Grafke/simplified-geometric-minimum-action-method-for-the-computation-of-instantons.html.
+).
 """
 function sgmam(
     sys::SgmamSystem,