Limit cycles for discontinuous generalized Lienard polynomial differential equations

Abstract

We divide $\mathbb{R}^2$ into sectors $S_1,\dots ,S_l$, with $l>1$ even, and define a discontinuous differential system such that in each sector, we have a smooth generalized Lienard polynomial differential equation $\ddot{x}+f_i(x)\dot{x} +g_i(x)=0$, $i=1, 2$ alternatively, where $f_i$ and $g_i$ are polynomials of degree n-1 and m respectively. Then we apply the averaging theory for first-order discontinuous differential systems to show that for any $n$ and $m$ there are non-smooth Lienard polynomial equations having at least max{n,m} limit cycles. Note that this number is independent of the number of sectors.

Keywords

• Limit cycles
• non-smooth Lienard systems
• averaging theory