Abstract. 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 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 non monotonicity window, which permits a better fitting between theoretical models and experimental 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.Key words. Piecewise linear systems; negative feedback circuits; periodic oscillations; Poincaré maps AMS subject classifications. 34, 921. Introduction. The investigation of the dynamical features of small regulatory units has been growing in interest for several decades, motivated by the necessity of controlling and predicting the global behaviors of biological organisms (see [39,41]). Many mathematical tools have been developed for this purpose, leading to several types of modeling frameworks in Systems and Synthetic Biology ([30, 24, 3, 42, 32]). Here we are interested in the link between continuous models (where the variations of concentrations in the molecules are represented by ordinary differential equations) and piecewise linear models as introduced by L. Glass in [14] (see [15, 17, 8]), where the equations combine piecewise constant production terms with linear degradation. The setting of piecewise linear models for biological systems has been widely studied during the last decades, as it provides useful analytical tools and explicit formulae of the solutions: for instance we refer the reader to [17,26,6,1,4,29,43] for an idea of the advances made since the work initiated by L. Glass, and to [10,5] for experimental validations of some of the results found theoretically. A useful direction in this field is to construct an approximation of a given system of ordinary differential equations by a piecewise linear system (in an appropriate state space grid) thus obtaining a simplified model for further theoretical