Constructing stellar objects with multiple necks

We discuss the construction of perfect fluid stellar objects having optical geometries with multiple necks corresponding to spatially closed unstable lightlike geodesics. We prove that there exist physically reasonable models with arbitrarily many necks. We also show how a first order phase transition can give rise to quite pronounced secondary double necks. The analysis is carried out using a modification of a recent dynamical systems formulation of the TOV equations due to Nilsson and Uggla. Our reformulation allows for a very general family of equations of state including, for example, phase transitions.