The shape of limit cycles for a class of quintic polynomial differential systems (Q2866703)

Consider the planar autonomous systems NEWLINE\[NEWLINE\dot x= \varepsilon^2\alpha_0 x-y+ P_5(x,y),\quad \dot y=x+ \varepsilon^2\alpha_0 y+ Q_5(x,y)\tag{1}NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\dot x= \varepsilon\alpha_0 x-y+\varepsilon P_5(x,y),\quad \dot y= x+\varepsilon^2 \alpha_0 y+\varepsilon Q_5(x,y),\tag{2}NEWLINE\]NEWLINE where \(P_5\) and \(Q_5\) are homogeneous polynomials of degree 5. The authors study the bifurcation of a limit cycle from the origin in system (1) and from a linear center in system (2). For this purpose they introduce polar coordinates, transform the systems into an abelian differential equation and apply second-order averaging and derive an asymptotic expression in \(\varepsilon\) for the amplitude of the bifurcating limit cycle.