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Periodic solutions of a singular equation with indefinite weight. (English) Zbl 1232.34064
The authors study the existence and uniqueness of $T$-periodic solutions for the equation $$ x''= \frac{a(t)}{x^3},$$ where $a$ is a $T$-periodic function given by $$ a(t) = a_+ \ \ \text{if} \ 0 \leq t < t_+, \ \ a(t) = -a_- \ \ \text{if} \ t_+ \leq t < T $$ with $a_+,a_- > 0.$ 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.

34C25Periodic solutions of ODE
34D20Stability of ODE