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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.

##### MSC:
 34C25 Periodic solutions to ordinary differential equations 34D20 Stability of solutions to ordinary differential equations
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