Fully characterizing self-oscillations in nonlinear oscillators demands knowledge both of the associated phase-space dynamics and the corresponding dual dynamics, which are visible in the stability diagrams of the system control parameter space. While phase-space dynamics have been extensively studied ever since the pioneering works by Poincaré and others well over a century ago, the associated description of stability diagrams had to wait for the advent of modern computers, particularly to characterize the intricate chaotic phases as well as the complex phases corresponding to periodic motions other than stationary or the simple nonstationary periodic oscillations appearing immediately after Hopf bifurcations. Experimentally, the determination of stability boundaries beyond the Hopf lines is a very difficult problem. Moreover, from a theoretical point of view, the determination of the frontiers between chaotic and periodic phases of oscillators is a complex task that needs to be addressed in all branches of the natural sciences. We observe experimentally and numerically a remarkable family of discontinuous spirals in the control parameter space of an electronic circuit.