Speaker
Description
The Delay Differential Equations (DDEs) are the class of differential equations where the derivative of the solution at the current time depends not only on the state of the system at that time, but also on the states in the past. This formulation leads to a (semi)dynamical system on an infinite dimensional phase space, similar to what happens in Partial Differential Equations (PDEs). Thus, finding analytical solutions is almost impossible and the numerical methods to obtain approximate solutions might be challenging. DDEs are commonly used in modelling real-life problems where reaction time is not instantaneous: robotics, biological systems, epidemics, and technological processes such as milling.
I will shortly discuss the method for obtaining not only numerical approximations of solutions to a rather general class of DDEs but also a validated (rigorous) algorithm to obtain bounds on the true solution in the vicinity of the approximation [1]. The current implementation in the C++ language by employing techniques of Automatic Differentiation is quite general and might be immediately applied to almost any system that is represented by a computer program (C++ subroutine). The core principles of the method are similar to those developed in CAPD library of validated solvers for ODEs, differential inclusions and dissipative PDEs [2]. The implementation is available as an Open-Source library on GitHub [3].
I am currently using this method in mathematically rigorous computer-assisted proofs of various dynamical phenomena such as the existence of periodic orbits in Mackey-Glass equation [1], existence of complicated solutions in discontinuous DDEs [4] or symbolic dynamics in ODE perturbed by a delayed term [5]. But the method might be applied to other problems, such as reachability or optimal control, with the explicit bounds on the error.
Szczelina, R.; Zgliczyński, P.; High-order Lohner-type algorithm for rigorous computation of Poincaré maps in systems of Delay Differential Equations with several delays, Foundations of Computational Mathematics, Volume 24, pages 1389–1454, (2024).
T. Kapela, M. Mrozek, D. Wilczak, P. Zgliczynski,CAPD::DynSys: a flexible C++ toolbox for rigorous numerical analysis of dynamical systems,Communications in Nonlinear Science and Numerical Simulation, Volume 101, October 2021, 105578.
github.com/robsontpm/capdDDEs, Accessed: 28-12-2024.
Gierzkiewicz, A.; Szczelina, R.; Sharkovskii theorem for Delay Differential Equations and other infinite dimensional dynamical systems., Submitted, arxiv.org/abs/2411.19190 (2024).
Benedek, G.; Krisztin, T; Szczelina, R.; Stable periodic orbits for Mackey-Glass type equations, Journal of Dynamics and Differential Equations (2024).
