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We present a two particle model to explain the mechanism that stabilizes abunch of positively charged ions in an "ion trap resonator" [Pedersen etal,Phys. Rev. Lett. 87 (2001) 055001]. The model decomposes the motion of the twoions into two mappings for the free motion in different parts of the trap andone for a compressing momentum kick. The ions' interaction is modelled by atime delay, which then changes the balance between adjacent momentum kicks.Through these mappings we identify the microscopic process that is responsiblefor synchronization and give the conditions for that regime.Comment: 12 pages, 9 figures; submitted to Phys Rev

A mapping approach to synchronization in the Zajfman trap : stability conditions and the synchronization mechanism