Time-delay Example 1

Example from the paper by L.A. Marquez-Martinez (C.R. Acad-.Sci. Paris, t. 329, serie 1, pages 545-550, 1999.)

Let us consider the following differential time-delay nonlinear system:

eqs={(Subscript[x, 1])'[t]-> (Subscript[x, 1])[t] u[t]+u[t-2],
(Subscript[x, 2])'[t]-> u[t]+u[t-1],
(Subscript[x, 3])'[t]-> u[t-1]-u[t-2]};
vars={(Subscript[x, 1])[t],(Subscript[x, 2])[t],(Subscript[x, 3])[t],u[t]};
TableForm[eqs]

Let us introduce the following Ore algebra A of differential time-delay operators

replA=ModelToReplacementRules[eqs,t];
A=OreAlgebraWithRelations[Der[t],S[-1][t],replA]

and the matrix R of ordinary differential time-delay operators, which defines the generic linearization of the above nonlinear system

MatrixForm[R=ToOrePolynomialD[eqs,vars,A]]

and the left A-module, finitely presented by R, defined by M = (A^(1×4))/(A^(1×3))R.

Let us compute the adjoint of R:

MatrixForm[Radj=Involution[R,A]]

Let us check whether or not M is torsion-free:

MatrixForm/@({Ann,Rp,Q}=Exti[Radj,A,1])

Since the first matrix is not identity, we deduce that M admits nontrivial torsion elements and thus the corresponding system admits autonomous elements τ[1], ...,  τ[4], defined by:

AutonomousElements[R,{(Subscript[dx, 1])[t],(Subscript[dx, 2])[t],(Subscript[dx, 3])[t],du[t]},τ,A,Relations->True]

We note that first three autonomous elements τ[1], τ[2], τ[3] are trivial. The only nontrivial autonomous element is τ[4] =-(Subscript[dx, 2])[-2+t]+(Subscript[dx, 2])[-1+t]-(Subscript[dx, 3])[-1+t]-(Subscript[dx, 3])[t], which satisfies (τ[4])'[t]==0. We find again the results of Example 1, studied in the paper L.A. Marquez-Martinez (C.R. Acad-.Sci. Paris, t. 329, serie 1, pages 545-550, 1999).