Nussbaum proposes a study of two related areas: (a) the dynamics in the large of nonlinear differential-delay equations with state dependent time lags and (b) questions about maps F:C--->C, where C is a closed cone in a Banach space and F is continuous, homogeneous of degree one and order-preserving in the partial ordering induced by C. The differential-delay equations to be studied are of the form
(1) ax'(t)=f(x(t), x(t- r_1), x(t- r_2), ...,x(t- r_n)),
where a>0 and r_j may depend on the history of the function x or r_j :=r_j (x(t)) may simply be a function of x(t). A special class of maps of the type described above in topic (b) is provided by "generalized max-plus operators", which Nussbaum and J. Mallet-Paret have shown are intimately related to equation (1). Even very special cases of equation (1) remain terra incognita. One may mention, for example, the equation
(2) ax'(t)= -x(t) - (k_1)x(t -r_1) -(k_2)x(t -r_2),
where a>0, k_j >0, r_j :=max(0, a_j +(c_j)x(t)), a_j >0 and c_j is unequal to 0. Areas which will be investigated include (a) periodic solutions of equation (1) (existence, stability and uniqueness), (b) singular perturbations, e.g., limiting behaviour of periodic solutions of eq. (1) as a--->0, (c) regularity of solutions (real analyticity? Gevrey class?), (d) Poincare-Bendixson theory for equation (1) and (e) studies of cone-preserving maps F:C--->C, both in their relation to differential-delay equations and to other applications.
A wide variety of of scientific phenomena, notably in mathematical biology, but also in optics (e.g., semiconductor lasers) and control theory, have been modeled by differential-delay equations with multiple and/or variable time lags. Although our main focus is theoretical, we believe that progress in understanding important model equations like eq. (1) and eq. (2) above will provide insight into equations which arise in applications. The importance of an underlying theoretical framework is already apparent in numerical studies of eq. (2) which suggest a wide variety of dynamical behaviour depending on the parameters. In equations from applications, which typically have many parameters, the need for a theoretical framework is even more acute.
