ParametrisationMethod

Parametrisation{ORD, NVAR, T}

A dense polynomial with a contiguous (FOM, ORD, L) coefficient array. Represents a parametrisation mapping from reduced coordinates and forcing variables to the full state.

• ORD: native order of the full ODE (1 for first‑order, 2 for second‑order).
• NVAR: total number of variables = reduced coordinates + forcing variables.
• T: numeric element type (e.g., ComplexF64).

Layout: coefficients[:, ord, l] is the full‑state vector (length FOM) for the ord-th time derivative of the l-th monomial coefficient.

ReducedDynamics{ROM, NVAR, T}

A dense polynomial whose coefficients are SVector{ROM, T}. Represents the reduced dynamics on a manifold of dimension ROM.

• ROM: dimension of the reduced state (first‑order system).
• NVAR: total number of variables = ROM + externalsystemsize.
• T: numeric type.

The polynomial is stored in poly, and external_system_size gives the number of forcing variables. The dynamics are: ż = R(z, r), where r are the forcing variables.

compute_higher_derivative_coefficients!(
	param_coeff, red_coeff, external_dynamics, superharmonic, global_index,
	generalised_eigenmodes, lower_order_couplings
) -> nothing

Compute the higher time‑derivative coefficients W^(j+1)[α] for j = 1 … ORD-1 using the superharmonic recurrence

W^(j+1)[α] = s · W^(j)[α]  +  Φ_master · R[α]  +  Φ_ext · e_dyn  +  ξ[j]

where:

• s = superharmonic is the frequency ⟨λ, α⟩,
• Φ = generalised_eigenmodes (FOM × NVAR) collects the right eigenmodes,
• R[α] = red_coeff[:, global_index] (ROM‑vector) contains the master‑mode reduced‑dynamics coefficients at the current monomial (already solved),
• e_dyn = external_dynamics (N_EXT‑vector) contains the known external dynamics at the current monomial,
• ξ[j] = lower_order_couplings[j] (FOM‑vector) contains the coupling from lower‑order monomials at derivative order j.

Modifies param_coeff in‑place. Does nothing when ORD = 1 (no higher derivatives exist for a first‑order ODE).

Arguments

• param_coeff :: AbstractArray{T, 3} — shape FOM × ORD × L; the coefficient tensor of the parametrisation polynomial.
• red_coeff :: AbstractMatrix{T} — shape ROM × L; master‑mode reduced‑dynamics coefficients.
• external_dynamics :: AbstractVector{T} — length N_EXT; known external dynamics at the current monomial.
• superharmonic :: T — scalar s = ⟨λ, α⟩.
• global_index :: Int — monomial index into the last axis of param_coeff and the last axis of red_coeff.
• generalised_eigenmodes :: AbstractMatrix{T} — shape FOM × NVAR; right generalised eigenvectors (master modes in columns 1:ROM, external modes in ROM+1:NVAR).
• lower_order_couplings :: AbstractVector{<:AbstractVector{T}} — length ORD; element j is a length‑FOM vector ξ[j] produced by LowerOrderCouplings.compute_lower_order_couplings.

create_parametrisation_method_objects(mset::MultiindexSet{NVAR}, ORD::Int, FOM::Int, ROM::Int, external_system_size::Int, ::Type{T}=Complex)

Create a consistent pair of polynomials:

• W: a Parametrisation{ORD, NVAR, T} with zero coefficients,
• R: a ReducedDynamics{ROM, NVAR, T} with zero coefficients.

Both polynomials share the same multiindex set mset and element type T. The total number of variables NVAR must satisfy NVAR == ROM + external_system_size. FOM is the full‑order dimension (size of the state vector). It is not stored but used to initialise the coefficient vectors correctly.

Arguments

• mset: multiindex set for NVAR variables.
• ORD: native order of the full ODE (1 or 2).
• FOM: dimension of the full‑order state in its native order.
• ROM: dimension of the reduced state.
• external_system_size: number of forcing variables (default 0).
• T: element type.