Reduction of Kinetic Equations to Li\'enard-Levinson-Smith Form: Counting Limit Cycles

We have presented an unified scheme to express a class of system of equations in two variables into a Li\'enard-Levinson-Smith (LLS) oscillator form. We have derived the condition for limit cycle with special reference to Rayleigh and Li\'enard systems for arbitrary polynomial functions of damping and restoring force. Krylov-Boguliubov (K-B) method is implemented to determine the maximum number of limit cycles admissible for a LLS oscillator atleast in the weak damping limit. Scheme is illustrated by a number of model systems with single cycle as well as the multiple cycle cases.