Particle amplitude growth due to single or repetitive resonance crossing

We study the problem of the growth of the amplitude of a particle executing betatron oscillations in a synchrotron or storage ring, due to the crossing of a resonance line. The resonance can be crossed either only once, or repeatedly. We treat only one degree of freedom in this paper, and assume the particles in the beam are independent. We consider some simple models of amplitude-dependent tunes, to discern the effects of detuning away from the resonance as the particle amplitude grows. The basic equation of motion we shall treat is the harmonic oscillator equation