Bifurcations of beam-beam like maps

Abstract

The bifurcations of a class of mappings including the beam-beam map are examined. These maps are asymptotically linear at infinity where they exhibit invariant curves and elliptic periodic points. The dynamical behaviour is radically different with respect to the Hénon-like polynomial maps whose stability boundary (dynamic aperture) is at a finite distance. Rather than the period-doubling bifurcations exhibited by the Hénon-like maps, we observe a systematic appearance of tangent bifurcations and in phase space one observes the disappearance of chains of islands born from the origin and coming from infinity. This behaviour has relevant consequences on the transport process.