# Nonlinear Resonance and Resonance Overlapping

## Abstract

Higher-order components in the magnetic field in a ring introduce nonlinear terms into the Hamiltonian and generate nonlinear resonances. This can lead to complicated motion for particles with large amplitudes of betatron oscillations. We derive the resonant structure in the phase space due to a sextupole magnet when the fractional part of the tune is close to \(\pm \frac{1}{3}\). For a Hamiltonian system with many resonances, they can interact with each other and lead to stochastic orbits in phase space. To understand this effect, we study a model called the *standard map*, that illustrates qualitative features of what can occur in a Hamiltonian system with many resonances. The impact on dynamics is similar whether originating from effects as diverse as nonlinear magnetic fields, RF cavities, space-charge forces among the charged particles in a bunch, or interactions between bunches.

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