Dimensional reduction of nonlinear delay differential equations with periodic coefficients using Chebyshev spectral collocation (Q840423)

The authors develop a numerical method for delay equations of the type \[ \dot x(t) = A(t)x(t) + B(t) x(t-\tau) + \varepsilon f(x(t), x(t-\tau), t) \] based on {\parindent6mm \begin{itemize}\item[1)] expansion in powers of \( \varepsilon\) up to order 3 \item[2)] representation of elements of the state space \(C^0([-\tau,0], {\mathbb{R}}^n)\) by their values at the Chebyshev points \item[3)] numerical representation of differential by a spectral differentiation matrix \( \bar D_{\tau}\). \end{itemize}} With matrices \(\bar A, \bar B\) derived from \(A\) and \(B\) above, the authors obtain a numerical approximation of the monodromy operator in the form \( U = (\bar D_{\tau} - \bar A)^{-1} \bar B\), and also a nonlinear map representing the time-\(\tau\)-map of the nonlinear evolution. In the critical case (nontrivial eigenvalues of \(U\) on the unit circle), a function whose graph is a center manifold is computed up to third order (using the invariance condition). In the non-critical case, a similar approach can yield an invariant manifold, provided conditions on internal non-resonance are observed. Finally, reduced equations are (numerically) obtained which allow to study bifurcations. The method is illustrated and compared to more standard numerical procedures in three examples.