Limit cycles as stationary states of an extended Harmonic Balance ansatz

A limit cycle is a self-sustained periodic motion appearing in autonomous ordinary differential equations. As the period of the limit cycle is a-priori unknown, it is challenging to find them as stationary states of a rotating ansatz. Correspondingly, their study commonly relies on brute-force time-evolution or on circumstantial evidence such as instabilities of fixed points. Alas, such approaches are unable to account for the coexistence of multiple solutions, as they rely on specific initial conditions. Here, we develop a multifrequency rotating ansatz with which we find limit cycles as stationary states. We demonstrate our approach and its performance in the simplest case of the Van der Pol oscillator. Moving beyond the simplest example, we show that our method can capture the coexistence of all fixed-point attractors and limit cycles in a modified nonlinear Van der Pol oscillator. Our results facilitate the systematic mapping of out-of-equilibrium phase diagrams, with implications across all fields of natural science.