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Twist solutions of a Hill’s equation with singular term. (English) Zbl 1016.34044

The author deals with \(2\pi\)-periodic twist solutions to the Brillouin equation \[ x''(t)+ w(t) x(t)= {1\over x(t)}, \] where \(w\) is a continuous \(2\pi\)-periodic function with an average value \(\gamma> 0\) over the period. The average value \(\gamma\) and the number \(\alpha= \gamma^{-1}\max w(t)- 1\) are considered as parameters of the problem. In the main theorem, the author determines a region of the parameters \(\gamma\) and such that a twist \(2\pi\)-periodic solution exists. As a consequence of the twist theorem of Moser, a twist solution is Lyapunov stable and in its neighborhood there exist quasiperiodic solutions and infinitely many subharmonic solutions with minimal period going to infinity. The method of the proof of the main result includes the tools of upper and lower solutions, topological degree and Birkhoff normal forms of area preserving maps.
Reviewer: Dmitrii Rachinskii

MSC:

34C25 Periodic solutions to ordinary differential equations
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