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 nega- tive 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 piecewise linear ones, each admitting 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. The interest of our approach lies in first, adopting a general class of functions, with a nonmonotonicity window, which permits a better fitting between theoretical models and experimen- tal data, and second, establishing a more accurate location for the periodic solution, which is useful for computational purposes in high dimensions. As an illustration, a model for the “repressilator” synthetic system is analyzed and compared to real data, and shown to admit a periodic orbit, for a range of activity functions.