High-dimensional Harmonic Balance Analysis for Second-order Delay-differential Equations

This paper demonstrates the utility of the high-dimensional harmonic balance (HDHB) method for locating limit cycles of second-order delay-differential equations (DDEs). A matrix version of the HDHB method for systems of DDEs is described in detail. The method has been successfully applied to capture the stable and/or unstable limit cycles in three different models: a machine tool vibration model, the sunflower equation and a circadian rhythm model. The results show excellent agreement with collocation and continuation-based solutions from DDE-BIFTOOL. The advantages of HDHB over the classical harmonic balance method are highlighted and discussed.