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A limit cycle is a self-sustained periodic motion appearing in autonomousordinary differential equations. As the period of the limit cycle is a-prioriunknown, it is challenging to find them as stationary states of a rotatingansatz. Correspondingly, their study commonly relies on brute-forcetime-evolution or on circumstantial evidence such as instabilities of fixedpoints. Alas, such approaches are unable to account for the coexistence ofmultiple solutions, as they rely on specific initial conditions. Here, wedevelop a multifrequency rotating ansatz with which we find limit cycles asstationary states. We demonstrate our approach and its performance in thesimplest case of the Van der Pol oscillator. Moving beyond the simplestexample, we show that our method can capture the coexistence of all fixed-pointattractors and limit cycles in a modified nonlinear Van der Pol oscillator. Ourresults facilitate the systematic mapping of out-of-equilibrium phase diagrams,with implications across all fields of natural science.

Limit cycles as stationary states of an extended Harmonic Balance ansatz