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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.

MSC:

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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