The existence of a stable limit cycle in the Liénard–Levinson–Smith (LLS) equation beyond the LLS theorem

Abstract

We have examined a class of Liénard–Levinson–Smith (LLS) oscillators having a stable limit cycle which demonstrates the fact that the LLS theorem can only be applied under an extended condition. The problem has been partly raised in a recent communication by Saha et al. (2020) (last paragraph of Sec 4.2.2). For the existence criteria we have provided a perturbative approach through Krylov–Bogolyubov averaging method to address this problem using the concept of energy consumption per cycle. We have also established the idea and its validity through proper numerical demonstration by considering a generalized model system. Such issues have potential utility in nonlinear vibration control.