Consider a linear system subject to non-linear feedback, i.e. a non linear element is present in the feedback loop. Assume that the element satisfies a sector condition , and (to keep things simple) that the open loop system is stable. Then the closed loop system is globally asymptotically stable if the Nyquist locus does not penetrate the circle having as diameter the segment located on the x-axis.
Haddad, Wassim M.; Chellaboina, VijaySekhar (2011). Nonlinear Dynamical Systems and Control: a Lyapunov-Based Approach. Princeton University Press.
ISBN9781400841042.