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

Abstract: 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 $\alpha \geq 1.9$. We complete the proof of Jones conjecture using global optimization methods, extended to treat infinite dimensional problems.

analysis of PDEsclassical analysis and ODEsdynamical systemsfunctional analysisnumerical analysis

Audience: researchers in the discipline

CRM CAMP (Computer-Assisted Mathematical Proofs) in Nonlinear Analysis

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