Limit cycles as stationary states of an extended harmonic balance ansatz

Abstract

A limit cycle is a self-sustained, periodic, isolated motion appearing in autonomous differential equations. As the period of a limit cycle is unknown, finding it as a stationary state of a rotating ansatz is challenging. Correspondingly, its study commonly relies on numerical methodologies (e.g., brute-force time evolution, and variational shooting methods) or circumstantial evidence such as instabilities of fixed points. Alas, such approaches are (i) , as they rely on specific initial conditions, and (ii) do not provide analytical intuition about the physical origin of the limit cycles. Here, we (I) develop a multifrequency rotating ansatz with which we (II) find limit cycles as stationary-state solutions via a semianalytical homotopy continuation. We demonstrate our approach and its performance on the Van der Pol oscillator. Moving beyond this simple example, we show that our method captures all coexisting 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 multiple fields of the natural sciences. Published by the American Physical Society 2024