Chaotic phase oscillation of a proton beam in a synchrotron

We investigate the chaotic phase oscillation of a proton beam in a cooler synchrotron. By using direct perturbation method, we construct the general solution of the 1st-order equation. It is demonstrated that the general solution is bounded under some initial and parameter conditions. From these conditions, we get a Melnikov function which predicts the existence of Smale-horseshoe chaos iff it has simple zeros. Our result under the 1st-order approximation is in good agreement with that in [H. Huang et al., Phys. Rev. E 48 (1993) 4678]. When the perturbation method is not suitable for the system, numerical simulation shows the system may present transient chaos before it goes into periodical oscillation; changing the damping parameter can result in or suppress stationary chaos.