| Abstract: |
| Delay differential equations (DDEs) are widely used in many applied fields to account for delayed responses of the modeled systems to either internal or external factors. In contrast to ordinary differential equations (ODEs), the phase space associated even with a scalar DDE is infinite-dimensional. Oftentimes, it is desirable to have low-dimensional ODE systems that capture qualitative features as well as approximate certain
quantitative aspects of the DDE dynamics.
In this talk, we present a new Galerkin scheme for general nonlinear DDEs. The main new ingredient is the use of a type of polynomials that are orthogonal under an inner product with a point mass. Rigorous convergence results will be presented, and the efficiency of the approach will be numerically illustrated on (i) characterizing the nature of Hopf bifurcations in a cloud-rain delay model as well as (ii) reproducing the chaotic attractors of some DDE models arising from climate dynamics consideration. The presentation is based on joint work with Mickael D. Chekroun (UCLA), Michael Ghil (UCLA \& ENS, France), Ilan Koren (Weizmann Institute, Israel), and Shouhong Wang (IUB). |
|