CRM CAMP in Nonlinear Analysis

Preuves mathématiques assistées par ordinateur en analyse non linéaire

12 janvier 2021 de 10 h 00 à 11 h 00 (heure de Montréal/HNE) Réunion Zoom

A computer assisted proof of chaos in a delayed perturbation of chaotic ODE

Séminaire par Robert Szczelina (Jagiellonian University, Poland)

We will discuss some recent developments to the Taylor method for forward in time rigorous integration of Delay Differential Equations (DDEs) with constant delays. We briefly discuss how to generalize method of the paper "Algorithm for rigorous integration of Delay Differential Equations and the computer-assisted proof of periodic orbits in the Mackey-Glass equation, Found. Comp. Math., 18 (6), 1299-1332, 2018" to incorporate multiple lags, multiple variables (systems of equations) and how to utilize "smoothing of solutions" to produce results of a far greater accuracy, especially when computing Poincaré maps between local sections. We will apply this method to validate some covering relations between carefully selected sets under Poincaré maps defined with a flow associated to a DDE. Together with standard topological arguments for compact maps it will prove existence of a chaotic dynamics, in particular the existence of infinite (countable) number of periodic orbits. The DDE under consideration is a toy example made by adding a delayed term to the Rössler ODE under parameters for which chaotic attractor exists. The delayed term is small in amplitude, but the lag time is macroscopic (not small). This is a joint work with Piotr Zgliczyński.