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Chebyshev collocation for linear, periodic ordinary and delay differential equations: a posteriori estimates

Ed Bueler

TL;DR

The paper develops a Chebyshev collocation framework for linear ODEs and linear, periodic DDEs with integer delays, and introduces a rigorous a posteriori error theory for both the collocation solution and the eigenvalues of the DDE monodromy operator. By embedding the DDE monodromy into a Hilbert-space setting and generalizing Bauer–Fike type perturbation results, it provides computable bounds that certify stability or instability of parameter regimes. The methodology extends to systems of DDEs, with detailed treatment of fundamental solution bounds, collocation for vector problems, and eigenvalue perturbation analyses, illustrated by examples such as a delayed, damped Mathieu equation. The work integrates spectral convergence, robust norm/evaluation techniques, and practical algorithms to produce reliable stability estimates for linear periodic ODEs and DDEs, including nontrivial delay structures. Overall, it offers a principled, numerically verifiable approach to DDE stability analysis via a posteriori spectral methods.

Abstract

We present a Chebyshev collocation method for linear ODE and DDE problems. We first give a posteriori estimates for the accuracy of the approximate solution of a scalar ODE initial value problem. Examples of the success of the estimate are given. For linear, periodic DDEs with integer delays we define and discuss the monodromy operator U as our main goal is reliable estimation of the stability of such DDEs. We prove a theorem which gives a posteriori estimates for eigenvalues of U, our main result. This result is based on a generalization to operators on Hilbert spaces of the Bauer-Fike theorem for (matrix) eigenvalue perturbation problems. We generalize these results to systems of DDEs. A delayed, damped Mathieu equation example is given. The computation of good bounds on ODE fundamental solutions is an important technical issue; an a posteriori method for such bounds is given. Certain technical issues are also addressed, namely the evaluation of polynomials and the estimation of L^\infty norms of analytic functions. Generalization to the non-integer delays case is also considered.

Chebyshev collocation for linear, periodic ordinary and delay differential equations: a posteriori estimates

TL;DR

The paper develops a Chebyshev collocation framework for linear ODEs and linear, periodic DDEs with integer delays, and introduces a rigorous a posteriori error theory for both the collocation solution and the eigenvalues of the DDE monodromy operator. By embedding the DDE monodromy into a Hilbert-space setting and generalizing Bauer–Fike type perturbation results, it provides computable bounds that certify stability or instability of parameter regimes. The methodology extends to systems of DDEs, with detailed treatment of fundamental solution bounds, collocation for vector problems, and eigenvalue perturbation analyses, illustrated by examples such as a delayed, damped Mathieu equation. The work integrates spectral convergence, robust norm/evaluation techniques, and practical algorithms to produce reliable stability estimates for linear periodic ODEs and DDEs, including nontrivial delay structures. Overall, it offers a principled, numerically verifiable approach to DDE stability analysis via a posteriori spectral methods.

Abstract

We present a Chebyshev collocation method for linear ODE and DDE problems. We first give a posteriori estimates for the accuracy of the approximate solution of a scalar ODE initial value problem. Examples of the success of the estimate are given. For linear, periodic DDEs with integer delays we define and discuss the monodromy operator U as our main goal is reliable estimation of the stability of such DDEs. We prove a theorem which gives a posteriori estimates for eigenvalues of U, our main result. This result is based on a generalization to operators on Hilbert spaces of the Bauer-Fike theorem for (matrix) eigenvalue perturbation problems. We generalize these results to systems of DDEs. A delayed, damped Mathieu equation example is given. The computation of good bounds on ODE fundamental solutions is an important technical issue; an a posteriori method for such bounds is given. Certain technical issues are also addressed, namely the evaluation of polynomials and the estimation of L^\infty norms of analytic functions. Generalization to the non-integer delays case is also considered.

Paper Structure

This paper contains 25 sections, 29 theorems, 134 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Interpolation and spectral differentiation at Chebyshev points
  3. A posteriori estimates for initial value problems
  4. An algorithm and error estimate
  5. Examples
  6. Proofs
  7. The monodromy operator for linear, periodic DDEs with integer delays
  8. The monodromy operator $U$ on $C(I)$
  9. A useful Hilbert space ${H^1}$
  10. The monodromy operator $\hat{U}$ on ${H^1}$
  11. Approximating the DDE monodromy operator and its eigenvalues
  12. Approximation of $U$
  13. Eigenvalue perturbation for diagonalizable operators on Hilbert spaces
  14. Approximating the eigenvalues of $U$
  15. An a posteriori theorem for the eigenvalues of $\hat{U}$
  16. ...and 10 more sections

Key Result

Theorem 1

Suppose $f$ is analytic in an open, simply-connected region $R$ such that $[-1,1]\subset R \subset \mathbb{C}$. Then there exist constants $C>0$ and $\rho>1$ independent of $N$ such that if $p=I_N f\in\mathcal{P}_N$ is the polynomial interpolant of $f$ at the $N+1$ Chebyshev collocation points $\mat Furthermore, suppose $E\subset R$ is an ellipse with foci $\pm 1$---see figure fig:ellipse below. I

Figures (16)

  • Figure 1: A stability chart for equation \ref{['introex']} with a (numerically-determined) stable point $(a,b)=(-1.1,1)$ indicated.
  • Figure 2: If $a=-1.1$ and $b=1$ then the largest three eigenvalues of the monodromy operator for equation \ref{['introex']} are proven to be within the given discs. Thus the parameter point $(a,b)$ is stable. (Unit circle and circle $r=\delta=0.2$ also shown.)
  • Figure 3: For DDE \ref{['introex']} with $(a,b)=(-1.1,1)$: The monodromy matrix is an $(N+1)\times (N+1)$ matrix approximation to the $\infty\times \infty$ monodromy operator. The radius found from theorem \ref{['thm:main']} of the error discs around the numerical eigenvalues of the monodromy matrix decays exponentially with $N$.
  • Figure 4: If $f$ is analytic in a region $R\subset \mathbb{C}$ which contains a "regularity ellipse" $E$ with foci $\pm 1$ and semiaxes $S,s$ then Chebyshev interpolation converges at exponential rate $\rho^{-1}$ where $\rho=S+s$. Note that $S^2=1+s^2$ for an ellipse with foci $\pm 1$.
  • Figure 5: Actual error $\|y-p\|_\infty$ and the estimate \ref{['specialone']} for the initial value problem in example \ref{['exampleone']}.
  • ...and 11 more figures

Theorems & Definitions (72)

  • Theorem 1: proven in Tadmor, in particular
  • Lemma 2: the collocation algorithm
  • Theorem 3
  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Lemma 4
  • proof
  • proof : Proof of theorem \ref{['thm:apostivp']}
  • ...and 62 more