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Zbl 1232.34064
Bravo, José Luis; Torres, Pedro J.
Periodic solutions of a singular equation with indefinite weight.
(English)
[J] Adv. Nonlinear Stud. 10, No. 4, 927-938 (2010). ISSN 1536-1365

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 2000:
*34C25 Periodic solutions of ODE
34D20 Lyapunov stability of ODE

Keywords: periodic solutions; singular equations; existence; uniqueness; linear stability

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