Phase portraits of quadratic systems with finite multiplicity three and a degenerate critical point at infinity
The following two theorems are proved. Theorem 1. Any quadratic system with three finite critical points (complex or coinciding) and an \(M^i_{1,3}\) type of infinite critical point can be transformed by a linear transformation into \[ \dot x=\lambda x+\mu y+ xy+ y^2,\quad \dot q=x+ y^2. \] There are 22 topologically different phase portraits; a limit cycle, if it exists, must be unique and hyperbolic. Theorem 2. Any quadratic system with three finite critical points and an \(M^i_{1,2}\) type of infinite critical point can be transformed into \[ \dot x= \lambda x+\mu y+ \gamma xy+\delta(x+ y^2),\quad \dot y= x+y^2,\;\lambda\in\{0,1\},\;\gamma\not\in\{0, 1\}. \] There are 119 topologically different phase portraits; a limit cycle, if it exists, must be unique and hyperbolic.
scientific article; zbMATH DE number 1143476
scientific article