Where the Li\'enard--Levinson--Smith (LLS) theorem cannot be applied for a generalised Li\'enard system

TL;DR

This paper investigates a class of Lie4nard--Levinson--Smith systems with stable limit cycles where the traditional LLS theorem does not apply, proposing a physical energy-based approach for analysis.

Contribution

It introduces an energy consumption per cycle method to analyze generalized Lie4nard systems where the LLS theorem fails, expanding understanding of nonlinear vibration control.

Findings

Identified cases where LLS theorem cannot be applied.

Developed an energy-based analysis approach.

Demonstrated the method on a generalized model system.

Abstract

We have examined a class of Li\'enard--Levinson--Smith (LLS) system having a stable limit cycle which demonstrates the case where the LLS theorem cannot be applied. The problem has been partly raised in a recent communication by Saha et al., CNSNS 85, 105234 (2020) (last para of sec 4.2.2). Here we have provided a physical approach to address this problem using the concept of energy consumption per cycle. We have elaborated the idea through proper demonstration by considering a generalized model system. Such issues have potential utility in nonlinear vibration control.