The constant-delay expansion conjecture for steady-state bifurcations

Let $u(t)=0$ be the steady state of a state-dependent delay differential equation, and suppose its expansion has terms through a given order involving only constant delays, with all remaining terms placed in a higher-order remainder. Standard normal form calculations for constant-delay DDEs may then be applied to the truncated constant-delay expansion. Constant-delay expansion conjecture. The local dynamics near the steady state $u(t)=0$ of the state-dependent delay equation are determined solely by the constant-delay expansion up to the given order. In other words, to study steady-state bifurcations, standard normal form calculations for constant-delay DDEs can be applied to the constant-delay expansion truncated to suitable order. This claim explains why the higher-order remainder can be omitted for the bifurcation calculations considered in the paper, but the source does not provide a rigorous resolution of the assertion.