This article is concerned with the design of synthetic central pattern generators (CPGs). Biological CPGs are neural circuits that determine a variety of rhythmic activities, including locomotion, in animals. A synthetic CPG is a network of dynamical elements (here called cells) properly coupled by various synapses to emulate rhythms produced by a biological CPG. We focus on CPGs for locomotion of quadrupeds and present our design approach, based on the principles of nonlinear dynamics, bifurcation theory, and parameter optimization. This approach lets us design the synthetic CPG with a set of desired rhythms and switch between them as the parameter representing the control actions from the brain is varied. The developed four-cell CPG can produce four distinct gaits: walk, trot, gallop, and bound, similar to the mouse locomotion. The robustness and adaptability of the network design principles are verified using different cell and synapse models.