Limit cycles in quadratic systems with a first-order weak focus
Consider the quadratic polynomial system \[ \begin{aligned} \dot x &= x^2+ xy+ y,\\ \dot y &= ax^2+ bxy+ cy^2- {1\over\varepsilon} x\end{aligned}\tag{\(*\)} \] under the conditions \[ c\in\Biggl({1\over 3}, 1\Biggr),\;c\neq{1\over 2},\;bc< 1,\;b> a+ c,\;2c< b+1, \] \[ 4a(c-1)> (b-1)^2,\quad 0<\varepsilon\ll 1. \] The author proves that under these assumptions system \((*)\) has three limit cycles, one limit cycle on the left of the straight line \(x=-1\), and two cycles on the right of this line. The proof is based on the reduction of \((*)\) to a Liénard system and studying the corresponding phase portrait.
scientific article; zbMATH DE number 5905507