Systèmes dynamiques
Dynamics of a Delay Differential Equation with two linearly state-dependent delays
Humphries, Tony
McGill University

Delay differential equations (DDEs) can be formulated as infinite dimensional dynamical systems, and the infinite dimensionality can lead to very interesting dynamics even in simple looking equations. We consider the model DDE $$\dot{u}(t)=-\gamma u(t)-\sum_{i=1}^2\kappa_i u(t-a_i-c_iu(t))$$ which has two linearly state-dependent delays, but apart from the state-dependency of the delays the DDE has no other nonlinearity. Nevertheless it displays very complex dynamics and bifurcation structures including bi-stability of periodic orbits, torus bifurcations and period doubling. We explore in depth the dynamics associated with the stable invariant tori  of the DDE including double Hopf bifurcations (and their unfolding), phase locking, Arnold tongues, and torus break up. The study also raises several numerical analysis issues in the computation of invariant manifolds and rotation numbers which we will touch upon.

Lundi, 17 juin, 11h00
Salle Des Plaines A