Pseudospectral reduction to compute Lyapunov exponents of delay differential equations (Q1670360)

The reduction of a Delay Differential Equation (DDE) to a system of Ordinary Differential Equations (ODEs) is based on the pseudospectral discretization introduced in [\textit{D. Breda} et al., SIAM J. Appl. Dyn. Syst. 15, No. 1, 1--23 (2016; Zbl 1352.34101)]. Here the authors consider the nonlinear autonomous DDE \[ x^{\prime}(t)=f(x_{t}), \] where, for \(\tau >0\) and \(X:=C([-\tau, 0]; \mathbb{R})\), \(f : X\rightarrow \mathbb{R}\) is smooth and \(x_{t}\in X\) denotes \(x_{t}(\theta):= x(t+\theta), \theta\in [-\tau, 0]\). By using the pseudospectral discretization, they get the reduced system of \(M+1\) ODEs \[ u^{\prime}_{0}(t)=f(U(t)) \] and \[ u^{\prime}_{i}(t)=\sum_{j=0}^{M} d_{i, j}u_{j}(t), \] \(i=1, \dots, M\), with initial conditions \(u_{i}(0)=\varphi (\theta_{i})\), \(i=0, 1, \dots, M\). The authors mention that a few papers address the question from a more numerical or computational point of view, but the methods proposed are usually difficult to replicate by non-experts and relevant algorithms and codes are rarely available or user-friendly. The authors' intention is to fill the gap between the need for off-the-shelf routines for those interested in the applications and rigorous methods. The authors consider the question of computing Lyapunov exponents of DDEs. To compute the Lyapunov exponents of ODEs the authors choose the discrete QR technique. They provide Matlab codes at the end of the paper as (i) \texttt{dqr}, implementing the discrete QR method for computing Lyapunov exponents of linear nonautonomus ODEs (ii) \texttt{solveDDE\_MG}, implementing the time-integration of nonlinear DDEs and (iii) \texttt{lrhs\_MG}, implementing the construction of the linear(ized) nonautonomous matrix. They mention that these are the first public and available codes for computing Lypaunov exponents of DDEs. The authors write that the methodology they propose has a potentially wide range of applications. The authors explain and use Matlab codes by illustrating some tests on three types of DDEs all with a single delay. It is mentioned that a rigorous proof of convergence of the computed Lyapunov exponents to the exact ones is not available, and out of the reach of the present work. In fact, the convergence of the approximating system of ODEs to the original DDE is still a subject of current research.