Rigorous theory for double Hopf (codimension-two) bifurcations in state-dependent delay DDEs

Establish a rigorous bifurcation theory for codimension-two double Hopf bifurcations in delay differential equations with state-dependent delays, including proofs of center manifold smoothness and derivation of normal forms that justify and organize the numerically observed dynamics for equations such as u(t) = −γ u(t) − κ1 u(t − a1 − c1 u(t)) − κ2 u(t − a2 − c2 u(t)).

Background

The authors demonstrate close agreement between numerically computed bifurcation structures and normal-form predictions obtained via heuristic expansions that freeze delays and reduce to constant-delay problems. They explicitly note that these results are obtained without a rigorous theory for double Hopf bifurcations in state-dependent delay systems.

A formal theory would require establishing sufficient smoothness of the center manifold in the state-dependent setting and deriving validated normal forms for the double Hopf case.