Seminar by Honghu Liu

Speaker

Honghu Liu (Virginia Tech)

Title

Galerkin Approximations of Nonlinear DDEs: Convergence Analysis and Applications to Noise-Driven Chaos

Date

  • November 28, 2023 16:00 CET+0100 (Europe/Rome)

  • November 28, 2023 10:00 EST-0500 (US/Eastern)

  • November 28, 2023 09:00 CST-0600 (US/Central)

  • November 28, 2023 07:00 PST-0800 (US/Pacific)

Abstract

Delay differential equations (DDEs) are widely used in many applied fields to account for delayed responses of the modeled systems to internal/external factors. In contrast to ODEs, the phase space associated even with a scalar DDE is infinite-dimensional. Oftentimes, it is desirable to have low-dimensional ODE systems to approximate the DDE dynamics. In this talk, we present a new Galerkin scheme for general nonlinear DDEs. The main ingredient is a type of polynomials that are orthogonal under an inner product with a point mass. The associated Galerkin scheme enjoys some nice properties that help reduce the derivation of the convergence results to basic functional analysis exercises. Analytic formulas are available within this approach, which help simplify the numerical treatment. We will also discuss further dimension reduction using the center manifold technique for DDE bifurcation analysis. It leads to a rigorous and computationally efficient way to approximate the Stuart-Landau normal forms for the considered DDEs. We will show how insights gained from such reduced systems can help design stochastic perturbations to induce chaos from the otherwise periodic DDE dynamics. The approach will be illustrated using a cloud-rain DDE model in the context of Hopf bifurcations and noise-driven chaos.

Recording

Watch the recording on our YouTube channel.