High-Order Lohner-Type Algorithm for Rigorous Computation of Poincaré Maps in Systems of Delay Differential Equations with Several Delays

• Published in: Foundations of computational mathematics Vol. 24; no. 4; pp. 1389 - 1454
• Main Authors: Szczelina, Robert, Zgliczyński, Piotr
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.08.2024; Springer; Springer Nature B.V
• Subjects: Algorithms; Applications of Mathematics; Chaos theory; Computation; Computer Science; Delay; Differential equations; Economics; Linear and Multilinear Algebras; Math Applications in Computer Science; Mathematics; Mathematics and Statistics; Matrix Theory; Numerical Analysis; Orbits; Poincare maps; Symbolic dynamics; Periodic orbits; Fixed-point index; 65Q20; Covering relations; 65G20; 34K38; Computer-assisted proofs; 34K13; Infinite-dimensional phase space; 34K23
• ISSN: 1615-3375; 1615-3383; 1615-3383
• DOI: 10.1007/s10208-023-09614-x

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 produce enclosures of solutions of a high order. We apply the topological techniques to prove various kinds of dynamical behaviour, 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).