CRM CAMP in Nonlinear Analysis

Computer-Assisted Mathematical Proofs in Nonlinear Analysis

May 11, 2021 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

A computer assisted proof of Wright's conjecture: counting and discounting slowly oscillating periodic solutions to a DDE

Seminar presented by Jonathan Jaquette (Boston University, USA)

A classical example of a nonlinear delay differential equation is Wright's equation: y'(t)=\alpha y(t−1)[1 + y(t)], considering \alpha>0 and y(t)>-1. This talk discusses two conjectures associated with this equation: Wright's conjecture, which states that the origin is the global attractor for all \alpha \in (0,\pi/2]; and Jones' conjecture, which states that there is a unique slowly oscillating periodic solution for \alpha>\pi/2. 

To prove Wright's conjecture our approach relies on a careful investigation of the neighborhood of the Hopf bifurcation occurring at \alpha=\pi/2. Using a rigorous numerical integrator we characterize slowly oscillating periodic solutions and calculate their stability, proving Jones' conjecture for \alpha \in [1.9,6.0] and thereby all \alpha \ge 1.9. We complete the proof of Jones conjecture using global optimization methods, extended to treat infinite dimensional problems.