This paper describes approximate return maps for the bound and pronk gaits in a simple quadruped model. Although expressions for the exact maps cannot be obtained, perturbation theory allows derivation of explicit expressions for approximate maps if particular parameters are assumed small. Equilibria of the approximate maps are presented as a function of system parameters and compare well with those of the exact maps. The simple model predicts that, for a given set of parameters, the pronk produces the greatest ground clearance, while the bound has the shortest gait period, thus allowing more frequent interaction of the feet with the ground to avoid obstacles or provide acceleration.