The stability of the equilibrium of reversible systems

Abstract:

In this paper, we consider the system \[ \dot x=a(t)y^{2m+1}+f_1(x,y,t),\quad \dot y=-b(t)x^{2n+1}+f_2(x,y,t),\] where $m,n\in \mathbf {Z}_+$, $m+n\ge 1$, $a(t)$ and $b(t)$ are continuous, even and 1-periodic in the time variable $t$; $f_1$ and $f_2$ are real analytic in a neighbourhood of the origin $(0,0)$ of $(x,y)$-plane and continuous 1-periodic in $t$. We also assume that the above system is reversible with respect to the involution $G\colon (x,y)\mapsto (-x,y)$. A sufficient and necessary condition for the stability in the Liapunov sense of the equilibrium $(x,y)=(0,0)$ is given.