Approximating strange attractors and Lyapunov exponents of delay differential equations using Galerkin projections

Delay differential equations (DDEs) are infinite-dimensional systems, so even a scalar, unforced nonlinear DDE can exhibit chaos. Lyapunov exponents are indicators of chaos and can be computed by comparing the evolution of infinitesimally close trajectories. We convert DDEs into partial differential equations with nonlinear boundary conditions, then into ordinary differential equations (ODEs) using the Galerkin projection. The solution of the resulting ODEs approximates that of the original DDE system; for smooth solutions, the error decreases exponentially as the number of terms used in the Galerkin approximation increases. Examples demonstrate that the strange attractors and Lyapunov exponents of chaotic DDE solutions can be reliably approximated by a smaller number of ODEs using the proposed approach compared to the standard method-of-lines approach, leading to faster convergence and improved computational efficiency.