Piecewise discretization of monodromy operators of delay equations on adapted meshes

TL;DR

This work addresses the challenge that standard pseudospectral discretizations of the monodromy operator for delay equations can lose convergence when the numerical periodic solution is computed on an adaptively refined mesh. The authors propose a piecewise discretization that includes the endpoints of the adapted period partition and, for the delay interval, appropriately aligns the grid, resulting in a piecewise operator whose eigenvalues (Floquet multipliers) converge with rates matching the underlying smoothness. They demonstrate both theoretical convergence consistency and extensive numerical validation across DDEs, renewal equations, and coupled systems, showing improved stability assessments and efficiency, particularly for strongly adapted solutions and oscillatory eigenfunctions. The approach enhances robustness of multiplier computations and broadens the applicability of collocation-based methods to a wider class of delay equations with nonuniform meshes. The practical impact lies in reliable local stability analysis for complex delayed dynamics using fewer nodes and preserving convergence when adaptation is essential.

Abstract

Periodic solutions of delay equations are usually approximated as continuous piecewise polynomials on meshes adapted to the solutions' profile. In practical computations this affects the regularity of the (coefficients of the) linearized system and, in turn, the effectiveness of assessing local stability by approximating the Floquet multipliers. To overcome this problem when computing multipliers by collocation, the discretization grid should include the piecewise adapted mesh of the computed periodic solution. By introducing a piecewise version of existing pseudospectral techniques, we explain why and show experimentally that this choice is essential in presence of either strong mesh adaptation or nontrivial multipliers whose eigenfunctions' profile is unrelated to that of the periodic solution.

Figures (14)

• Figure 1: Periodic solutions (rescaled to period $1$) of \ref{['logisticdde']} computed by DDE-BIFTOOL with $L_{\text{DB}} = 30$ and $m_{\text{DB}} = 6$.
• Figure 2: Absolute errors of eigTMN, varying $M = N$, on the trivial (A--D) and dominant nontrivial (E--H) multipliers of \ref{['logisticdde']} linearized around the solutions computed by DDE-BIFTOOL. The gray horizontal lines show DDE-BIFTOOL's errors on the same multipliers. The reference values for the nontrivial multipliers are computed by DDE-BIFTOOL with $L_{\text{DB}} = 60$ and $m_{\text{DB}} = 10$. eigTMN's errors eventually decay with infinite order with barriers comparable to DDE-BIFTOOL's errors, except for the nontrivial multiplier with $r = 3$ (H), which seems to need even higher $M=N$. For the trivial multiplier, as $r$ increases (D), the errors initially rise exponentially (exceeding $10^7$ for $r = 3$), complicating the choice of $M = N$.
• Figure 3: Partitions of $[0, \omega]$ for the periodic solutions of \ref{['fig:logisticddesol']}, showing mesh adaptation as performed by DDE-BIFTOOL. The vertical lines show the uniform partition.
• Figure 4: Example collocation grid with $\omega > \tau$, $L=4$ and $M = 3$. Ticks mark $t_{i}$ and $t_{i} - \omega$, the cross marks $-\tau$, dots mark the grid points.
• Figure 5: Absolute errors of eigTMNpw on the dominant multiplier of \ref{['tent']}, whose coefficient is not differentiable at $1$. The reference value is computed by eigTMNpw with $L_{\text{pw}} = 2$ and $M_{\text{pw}} = 120$. When the mesh includes $1$ ($L_{\text{pw}} = 2$) the convergence order is infinite, otherwise ($L_{\text{pw}} = 1$) it is finite (precisely $2$).

Theorems & Definitions (2)

• Remark 2.1
• Remark 5.1