High-order Lohner-type algorithm for rigorous computation of Poincaré maps in systems of Delay Differential Equations with several delays

TL;DR

The paper develops a high‑order Lohner‑type algorithm for rigorous time‑integration of delay differential equations with multiple delays, embedding the solutions in a piecewise Taylor framework and exploiting solution smoothing to obtain high‑order enclosures. It extends the Lohner approach with a generalized phase‑space representation, nonuniform jet orders, and efficient handling of multiple delays, enabling rigorous computation of Poincaré maps. Leveraging Fixed Point Index on ANRs and covering relations extended to infinite‑dimensional spaces, the authors prove the existence of (apparently) unstable periodic orbits in Mackey‑Glass and persistence of symbolic dynamics in a delay‑perturbed Rössler system. Benchmarks show substantially tighter enclosures than prior methods, supporting computer‑assisted proofs of complex dynamics in both finite‑ and infinite‑dimensional DDEs, with potential to certify chaotic behavior in broader classes of DDEs. The work integrates advanced numerical rigor, topological methods, and application‑driven proofs, contributing a robust toolkit for dynamical analysis of multi‑delay systems.”

Abstract

We present a Lohner-type algorithm for rigorous integration of systems of Delay Differential Equations (DDEs) with multiple delays and its application in computation of Poincaré maps to study the dynamics of some bounded, eternal solutions. The algorithm is based on a piecewise Taylor representation of the solutions in the phase-space and it exploits the smoothing of solutions occurring in DDEs to produces enclosures of solutions of a high order. We apply the topological techniques to prove various kinds of dynamical behavior, for example, existence of (apparently) unstable periodic orbits in Mackey-Glass Equation (in the regime of parameters where chaos is numerically observed) and persistence of symbolic dynamics in a delay-perturbed chaotic ODE (the Rössler system).

Figures (6)

• Figure 1: An example of a covering relation $N \stackrel{P}{\Longrightarrow} N$ on an h-set with tail $N = (N_1, |N_2|)$, $u_{N_1} = 1, s_{N_1} = 1$. The tail $|N_2|$ is closed and convex in a potentially infinite dimensional space. The legend is as follows: the set $|N|$ is the parallelepiped in the middle, whereas its image $P(|N|)$ is stretched across $N$. The finite dimensional part is drawn in (x,y)-plane (width and height of the page), where the tail is drawn in z-coordinate (depth). The yellow thick line is one copy of the set $|N_2|$ (the tail part), blue thick lines mark the set $N_1^+$ (the ,,entrance set" of the finite dimensional part of $N$), red thick lines mark the set $N_1^-$ (the ,,exit set" of the finite dimensional part of $N$), light-blue and light-red polygons mark the entrance set $N_1^+ \times |N_2|$ and the exit set $N_1^- \times |N_2|$, respectively. The grey planes denote the boundary of the strip $\mathbb{R}^{d_{N_1}} \times |N_2|$ - the image of $|N|$ under $P$ is forbidden to extend beyond those planes in z-coordinate due to the condition (C0). The set $P(|N|)$ does not ,,touch" the entrance set $N_1^+ \times |N_2|$ - condition (C3) and the exit set $N_1^+ \times |N_2|$ is mapped outside $|N|$ (red polytopes on left and right part of the picture) - condition (C2). Please note, that the image $P(|N|)$ is allowed to touch the boundary $N_1 \times \mathop{\mathrm{\partial}}\nolimits |N_2|$ (place marked with a black arrow) as long as it does not go beyond the grey planes. It is also allowed to bend in the stable direction of the finite dimensional part outside the strip bounded by yellow hyperplanes (see right part of the picture). It is easy to see, that the map $P$ can be homotopied, with a straight line homotopy fulfilling condition (C0), to a map $(x, y, r) \mapsto (2\cdot x, 0, \bar{r})$, where $\bar{r} \in |N_2|$ (up to the coordinate change $c_N$) - condition (C4).
• Figure 2: Numerically observed attractor in the Rössler ODE for classical values of parameters: $a = b = 0.2$, $c = 5.7$. Picture generated by integrating forward in time single trajectory for a long time.
• Figure 3: The numerically observed attractors for the system studied in Theorem \ref{['thm:rossler-delayed']}. The cases (a)-(c) are shown from left to right, respectively. The grey attractor is the very long trajectory $v(t)$ obtained for a single constant initial function. The section $S_0$, represented as a green rectangle on the picture, spans in fact across the space $\mathbb{R}^3$, as can bee seen by the red to blue region that shows the segments of $v$ which lie on the section $S_0$, i.e. the set $\{ v_t : \pi_x(v(t)) = 0 \}$. The colours are assigned with ascending $\pi_y v(t)$ value. Those segments are used to define the $W_u$ coordinate in the set $X(A, \Xi)$.
• Figure 4: The rigorous estimates obtained in the computer assisted part of the proof of Theorem \ref{['thm:rossler-delayed']}. The cases (a)-(c) are presented top to bottom, respectively. The left picture shows the representation of the computer assisted proof of the trapping region $X_A$. The set is divided into 200 pieces $X_{A,i}$ along the $W_u$ direction, each piece is coloured according to ascending number. Then for each piece $X_{A,i}$ the image $P(X_{A,i})$ is computed and drawn in the same colour (but with increased intensity). The dimension of the boxes in the $y$ coordinate represents the hull of the nominally stable part of the set $P(X_{A,i})$, i.e. the interval $I_i = [y_{lo}, y_{up}]$ such that all $\pi_{A_j} P_c(X) \subset I_i$, for $j \in \{3, \ldots, M\}$ and $\pi_{\Xi_j} (P_c(X)) \subset I$ for $j \in \{ 1, \ldots, p \cdot d \}$. Obviously, each $I_i \subset B_1\left(0, \max (P(r_A), P(r_\Xi)) \right)$. A clear evidence of the Smale horseshoe-like dynamics can be seen in the picture, as the box is folding on itself under the map $P$. On the right picture one there are represented the sets $X_1$ (light red, with red and yellow borders) and $X_2$ (light blue, with blue and purple borders). The images of the borders under the map $P^2$ are presented as lines (in fact thin boxes) in the grey area outside $X_1 \cup X_2$. It is evident that $P(W_{1,l})$ (red) and $P(W_{2,r})$ (purple) are both mapped to the left of both sets and $P(W_{1,r})$ (yellow) and $P(W_{2,l})$ (blue) are mapped to the right. Therefore condition (CC2A) is satisfied between the sets $X_1$ and any of $X_i$'s, and condition (CC2B) between $X_2$ and any $X_i$, $i \in \{ 1, 2 \}$. Please consult online version of the plots for better quality.
• Figure 5: (a) The apparently chaotic attractor of the Mackey-Glass equation \ref{['eq:mg']} for the classical parameter values $\tau = 2$, $n = 9.65$, $\beta = 2$, $\gamma = 1$mackey-glass. The attractor is drawn for a single very long solution, whose time-delay embedding coordinates $(x(t), x(t-\tau))$ are shown in the picture. (b) The representation of the attractor drawn in the coordinates $\left(x_n(0), x_{n+1}(0)\right)$, where $x_{n+1} = P(x_n)$, $x_n, x_{n+1} \in C([-\tau, 0], \mathbb{R})$. The map $P$ is constructed on the section $S = \{ x : x(t) = x(t-\tau), x(t) < 0.96 \}$, see Figure. 13 in mackey-glass-scholarpedia. The periodic points $T^1, T^2, T^4$ of respective periods 1, 2 and 4 for map $P$ are drawn in colors blue, green, red. (c) The same solutions are drawn in the time-delay embedding of the attractor and (d) as the solutions over time long enough to contain basic periods of all presented solutions.

Theorems & Definitions (50)

• Proposition 1
• Remark 2
• Definition 1
• Proposition 3
• Proposition 4
• Remark 5: On treating jets as vectors and vice-versa
• Lemma 6
• Definition 2
• Lemma 7
• Lemma 8