Qualitative analysis of the (N + 1)-body ring problem

In this paper we present a complete study of the (N + 1)-body ring problem. In particular, we review and describe the evolution of the equilibrium points, their stability, their bifurcations, the zero velocity curves and we provide new techniques that give new views to this classical problem. Some of these techniques are the OFLI2 (a Chaos Indicator given in [Barrio R. Sensitivity tools vs. Poincaré sections. Chaos, Solitons & Fractals 2005;25(3):711–26; Barrio R. Painting chaos: a gallery of sensitivity plots of classical problems. Int J Bifur Chaos Appl Sci Eng [in press]]) and the Crash Test [Nagler J. Crash test for the restricted three-body problem. Phys Rev E (3) 2005;71(2):026227, 11; Nagler J. Crash test for the Copenhagen problem. Phys Rev E (3) 2004;69(6):066218, 6]. With the OFLI2 we have studied the chaoticity of the orbits and with the Crash Test we have classified the orbits as bounded, escape or collisions. Finally, we have performed a systematic search of symmetric periodic orbits of the system, locating much more orbits that in previous studies of other authors.