Existence of limit cycles for quadratic systems with one infinite singular point
The author discusses the system \[ x=-y+ax^ 2-xy,\quad y=x+\delta y+\ell x^ 2+mxy+ny^ 2, \] with the condition \((m-a)^ 2<4\ell (n+1)\) or \(a=0\) by changing it into a Liénard equation on the region \(x>1:\) \(\dot x=y-F(x)\), \(\dot y=-g(x)\), where \[ F(x)=\int^{x}_{0}f(x)dx,\quad f(x)=\psi (x)(1+x)^{n-2},\quad g(x)=x\phi (x)(1+x)^{2n-3}, \] \[ \psi (x)=-(a+m+2na)x^ 2-(\delta +2a+m)x-\delta,\quad \phi (x)=(a^ 2n+\ell +am)x^ 3+(am+1+a\delta +2\ell)x^ 2+(\ell +2+a\delta)x+1. \] He obtains theorems on the existence and uniqueness of limit cycles under some additional conditions.
scientific article; zbMATH DE number 4134475