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Boundary-value problems of functional differential equations with state-dependent delays

Alessia andò, Jan Sieber

TL;DR

This work addresses the numerical approximation of time-periodic solutions to functional differential equations with state-dependent delays, for which standard differentiability requirements fail. By introducing a mild differentiability framework and reformulating periodic BVPs as fixed-point problems in spaces of 1-periodic functions, the authors prove that piecewise polynomial collocation converges with rate $O(L^{-\min\{\ell_{\max},m\}})$ to a locally unique infinite-dimensional solution. The analysis hinges on a discretized fixed-point map $\Phi_L=\mathcal{L}\mathcal{P}_L g$, a consistency estimate, and a contraction argument that bound nonlinear terms; the results align with convergence rates known for smooth cases despite the weaker regularity. The findings provide a rigorous justification for using collocation-based solvers (e.g., in DDE-Biftool) to continue periodic orbits in FDEs with state-dependent delays and open avenues for extending to spectral methods and more general delay equations, including neutral types. The illustrative Hopf-bifurcation example demonstrates the practical applicability and reliability of the proposed approach.

Abstract

We prove convergence of piecewise polynomial collocation methods applied to periodic boundary value problems for functional differential equations with state-dependent delays. The state dependence of the delays leads to nonlinearities that are not locally Lipschitz continuous preventing the direct application of general abstract discretization theoretic frameworks. We employ a weaker form of differentiability, which we call mild differentiability, to prove that a locally unique solution of the functional differential equation is approximated by the solution of the discretized problem with the expected order.

Boundary-value problems of functional differential equations with state-dependent delays

TL;DR

This work addresses the numerical approximation of time-periodic solutions to functional differential equations with state-dependent delays, for which standard differentiability requirements fail. By introducing a mild differentiability framework and reformulating periodic BVPs as fixed-point problems in spaces of 1-periodic functions, the authors prove that piecewise polynomial collocation converges with rate to a locally unique infinite-dimensional solution. The analysis hinges on a discretized fixed-point map , a consistency estimate, and a contraction argument that bound nonlinear terms; the results align with convergence rates known for smooth cases despite the weaker regularity. The findings provide a rigorous justification for using collocation-based solvers (e.g., in DDE-Biftool) to continue periodic orbits in FDEs with state-dependent delays and open avenues for extending to spectral methods and more general delay equations, including neutral types. The illustrative Hopf-bifurcation example demonstrates the practical applicability and reliability of the proposed approach.

Abstract

We prove convergence of piecewise polynomial collocation methods applied to periodic boundary value problems for functional differential equations with state-dependent delays. The state dependence of the delays leads to nonlinearities that are not locally Lipschitz continuous preventing the direct application of general abstract discretization theoretic frameworks. We employ a weaker form of differentiability, which we call mild differentiability, to prove that a locally unique solution of the functional differential equation is approximated by the solution of the discretized problem with the expected order.

Paper Structure

This paper contains 26 sections, 12 theorems, 66 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Functional differential equations (FDEs)
  3. Collocation for FDEs
  4. Lack of continuous differentiability of the nonlinearity
  5. Convergence of numerical discretization methods for BVPs
  6. Main results
  7. Periodic BVPs for FDEs and mild differentiability
  8. Convergence of solution of discretized BVP
  9. Proof through equivalent fixed-point problem
  10. Illustrative example of BVP
  11. Fixed-point problem equivalent to periodic BVP and relevant notation
  12. Notation for function spaces and norms
  13. Linear and nonlinear part of fixed-point problem $x=\Phi(x)$
  14. Discretized fixed-point problem
  15. Example
  16. ...and 11 more sections

Key Result

Theorem 2.3

Under Assumption res:ass the discretized BVP res:bvp:disc with $\max\{t_i-t_{i-1}\}\leq C_\mathrm{msh}/L$ has a locally unique solution $x_L=(y^L,T^L,p^L)$ near $x^*=(y^*,T^*,p^*)$ for all sufficiently large $L$, satisfying

Figures (1)

  • Figure 7.1: Left: maximum among the errors computed at $10001$ equidistant points in $[0,T]$ for different values of $m$ and $L$, compared to straight dashed lines having slope $m$. Right: periodic solution computed with $L=10$ and $m=5$, rescaled so as to have period 1.

Theorems & Definitions (20)

  • Definition 2.1: Mild differentiability
  • Theorem 2.3: Convergence of discretization
  • Lemma 3.1: Equivalence of discretized fixed point problem
  • Remark 3.2: Discretized solution space
  • Lemma 4.1: Extended differentiability of nonlinearity with time shift
  • Corollary 4.2: Extension of finite-difference limit
  • Corollary 5.3: Regularity of solution of fixed-point problem \ref{['Phi1']}
  • Lemma 6.1: Consistency of derivative of $\Phi_L$
  • proof
  • Remark 6.2: Sharper estimate if $G$ is mildly differentiable twice
  • ...and 10 more