*(English)*Zbl 1232.34064

The authors study the existence and uniqueness of $T$-periodic solutions for the equation

where $a$ is a $T$-periodic function given by

with ${a}_{+},{a}_{-}>0\xb7$ These problems arise in different physical situations such as in the stabilization of matter-wave breathers in Bose-Einstein condensates, in the propagation of guided waves in optical fibers and in the electromagnetic trapping of a neutral atom near a charged wire. If the parameters ${a}_{+},{a}_{-}$ are fixed, and $T:={t}_{+}+{t}_{-},$ an interesting question is how to control the switching times ${t}_{-},{t}_{+}$ in order to get periodic states with a particular amplitude. This question is studied in the paper as well as the stability properties (in the linear sense) of the $T$-periodic solutions.