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

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