## Stability of the phase motion in race-track microtrons.(English)Zbl 1376.78006

Summary: We model the phase oscillations of electrons in race-track microtrons by means of an area preserving map with a fixed point at the origin, which represents the synchronous trajectory of a reference particle in the beam. We study the nonlinear stability of the origin in terms of the synchronous phase – the phase of the synchronous particle at the injection. We estimate the size and shape of the stability domain around the origin, whose main connected component is enclosed by an invariant curve. We describe the evolution of the stability domain as the synchronous phase varies. We also clarify the role of the stable and unstable invariant curves of some hyperbolic (fixed or periodic) points.

### MSC:

 78A55 Technical applications of optics and electromagnetic theory 37N20 Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics) 37M05 Simulation of dynamical systems 39A30 Stability theory for difference equations
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### References:

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