The purpose of this paper is to show how the stability of limit cycles in third and higher-order systems having non-linearities with a real describing function can be ascertained from the negative slope of the describing function at its point of intersection with a locus dependent on the coefficients of the numerator and denominator polynomials of the transfer function. It is also shown that the amplitude of stable oscillations depends both on the non-linearity and the linear part of the plant and can be determined from the point of intersection whereas the frequency is independent of the non-linearity and can be computed from the coefficients of the linear plant alone. In the case of an all-pass transfer function and for a specific non-linearity the amplitude depends on the gain but is independent of the coefficients of the linear plant. The maximum gain giving stable limit cycles can be related to the non-linearity and the coefficients of the linear plant.