Periodic solutions of certain hybrid delay-differential equations and their corresponding difference equations

In this paper we use a method due to Carvalho [L.A.V. Carvalho, On a method to investigate bifurcation of periodic solution in retarded differential equations, J. Difference Equ. Appl. 4 (1998) 17–27] to obtain conditions for the existence of nonconstant periodic solutions of certain systems of hybrid delay-differential equations. We first deal with a scalar equation of Lotka-Valterra type, followed by a system of 2 equations in 2 unknowns that could model the interactions of 2 identical neurons. It will be seen that such solutions are determined by solutions of corresponding difference equations. Another paper in which this method is used is by Cook and Ladeira [K.L. Cook, L.A.C. Laderia, Applying Carvalho’s method to find periodic solutions of difference equations, J. Difference Equ. Appl. 2 (1996) 105–115. [2]].