Periodic solutions of the Brillouin electron beam focusing equation (Q380181)

The authors study the existence of \(2\pi\)-periodic solutions for a scalar second order equation of the type NEWLINE\[NEWLINE \ddot x+b(1+\cos t)x=\frac{1}{x}\,, NEWLINE\]NEWLINE where \(b\) is a positive constant. The difficulty lies in the fact that the function \(a(t)=b(1+\cos t)\) vanishes at some points. In the previous literature, existence was established in cases when \(b\) lies in the first interval of stability of the associated Mathieu equation \(\ddot x+b(1+\cos t)x=0\).NEWLINENEWLINEIn this paper, it is shown that there exists a \(2\pi\)-periodic solution for some \(b\) which lies in the second interval of stability. Precisely, Theorem 1 states that, if \(b\in[0.4705, 0.59165]\), there is at least one \(2\pi\)-periodic solution.NEWLINENEWLINEThe proof uses phase-plane analysis, together with the Poincaré-Bohl fixed point theorem.