Reduction of Kinetic Equations to Liénard–Levinson–Smith Form: Counting Limit Cycles

Abstract

We have presented an unified scheme to express a class of system of equations in two variables into a Liénard–Levinson–Smith (LLS) oscillator form. We have derived the condition for limit cycle with special reference to Rayleigh and Liénard 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.

Appendix: Lotka–Volterra System

To obtain the LLS form of Lotka–Volterra System [3,4,5], let us set $z=\delta x+\beta y$ then ${\dot{z}}=\alpha \delta x-\beta \gamma y=u$ $\implies$ $x=\frac{{\dot{z}}+\gamma z}{(\alpha +\gamma )\delta }$ and $y=\frac{-{\dot{z}}+\alpha z}{(\alpha +\gamma )\beta }$. After taking t derivative upon ${\dot{z}}$ one can have,

$$\begin{aligned} \ddot{z} =(\alpha -\gamma ){\dot{z}}+\alpha \gamma z +\frac{{\dot{z}}^2}{\alpha +\gamma }+\frac{\gamma - \alpha }{\alpha +\gamma } z {\dot{z}}-\frac{\alpha \gamma }{\alpha +\gamma } z^2. \end{aligned}$$

The fixed point (0, 0) gives a saddle solution which is not of any interest in the present context. Choosing the remaining non-zero fixed point for further investigations, and after taking perturbation $z=\xi +z_s$ around the fixed point $z_s=\alpha +\gamma =\delta x_s+\beta y_s \ne 0$, one can get the LLS form with $F(\xi ,{\dot{\xi }})=a_1 \xi +a_2 {\dot{\xi }}$ with $a_1=\frac{\alpha -\gamma }{\alpha +\gamma }$ and $a_2= - \frac{1}{\alpha +\gamma }$. It is to be noted that $G(\xi )$ contains nonlinearity with $G(\xi )=\omega ^2 \xi +a_3 \xi ^2$ where $\omega =\sqrt{\alpha \gamma }=Im(\lambda )$($+ve$ sense) and $a_3=\frac{\alpha \gamma }{\alpha +\gamma }$. After introducing a small parameter $\epsilon _1$ (say) in the constants, $a_i, b_i$ such that $a_i=\epsilon _1 b_i, i=1,2,3$ the above equation reduces to $\ddot{\xi }+\epsilon _1 (b_1 \xi +b_2 {\dot{\xi }}) {\dot{\xi }}+\omega ^2 \xi +\epsilon _1 b_3 \xi ^2=0$.