Existence and uniqueness of elliptic periodic solutions of the Brillouin electron beam focusing system. (English) Zbl 0966.34038

The ordinary differential equation \[ x''(t) = a(1 + \cos(t))x(t) - \frac{b}{x(t)} \tag{1} \] where \(a, b\) are positive constants is considered. Equation (1) describes the motion of a magnetically focused axially symmetric electron beam under the influence of a Brillouin flow. It is clear that from the mathematical point of view equation (1) is a singular perturbation of the Mathieu equation. The existence of several types of positive \(2 \phi\)-periodic solutions to equation (1) is proved when \(a\leq \frac {1}{16}\) by using an analysis of the phase plane.


34C25 Periodic solutions to ordinary differential equations
34C60 Qualitative investigation and simulation of ordinary differential equation models
74J30 Nonlinear waves in solid mechanics
81V10 Electromagnetic interaction; quantum electrodynamics
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