Leonhard Lücken:

We consider scalar delay dierential equations (DDEs) with long delay (see equation (1)).

In some cases, there is a correspondence of solutions of (1) and solutions of spatially extended systems which are derived by formal arguments. This perspective to DDEs with long delay was firstly described and investigated by Kashchenko [1] and Arecchi et al. [2].
I will introduce the asymptotic continuous spectrum (PCS) of (1) and explain how an amplitude equation describes the dynamics of (1) close to a destabilization which is induced by the PCS. For the case of cubic nonlinearities, I will present an error estimate for the deviation of the original solution from its formal approximation [3]. This is a first step to put the spatio-temporal representation of DDEs on solid grounds.

[1] S. A. Kashchenko. "Normalization Techniques as Applied to the Investigation of Dynamics of Difference-Differential Equations with a Small Parameter Multiplying the Derivative." Dier. Uravn 25 (1989): 1448-1451.

[2] F. T. Arecchi , G. Giacomelli, A. Lapucci, and R. Meucci. "Two-dimensional representation of a delayed dynamical system." Physical Review A 45, no. 7 (1992): R4225-R4228.

[3] S. Yanchuk, L. Lücken, M. Wolfrum, and A. Mielke. "Spectrum and amplitude equations for scalar delay-differential equations with large delay" in preparation