Numerical computation of unstable invariant tori in state-dependent delay DDEs

Develop numerical continuation and computation methods for unstable invariant tori produced by torus bifurcations in delay differential equations with state-dependent delays, specifically for the two-delay model u(t) = −γ u(t) − κ1 u(t − a1 − c1 u(t)) − κ2 u(t − a2 − c2 u(t)).

Background

The paper computes and visualizes stable quasi-periodic and phase-locked tori via initial value simulations and unstable-manifold techniques for embedded periodic orbits. However, it explicitly notes the lack of a method to compute unstable tori directly.

This limitation restricts comprehensive bifurcation analysis and characterization of torus dynamics in state-dependent delay systems, motivating development of new numerical algorithms.