Negative feedback circuits are a recurrent motif in regulatory biological networks, strongly linked to the emergence of oscillatory behavior. The theoretical analysis of the existence of oscillations is a difficult problem and typically involves many constraints on the monotonicity of the activity functions. Here, we study the occurrence of periodic solutions in an n-dimensional class of negative feedback systems defined by smooth vector fields with a window of not necessarily monotonic activity. Our method consists in circumscribing the smooth system by two hybrid systems of a specific type, called piecewise linear systems. We can show that each of these hybrid systems has a periodic solution. It can then be shown that the smooth negative feedback system also has a periodic orbit, inscribed in the topological solid torus constructed from the two piecewise linear orbits. As an illustration, a model for the “Repressilator” (a synthetic biological system of three genes) is analyzed and compared to real data, and shown to admit a periodic orbit, for a range of activity functions.