On the number of limit cycles of a \(\mathbb{Z}_4\)-equivariant quintic near-Hamiltonian system (Q889458): Difference between revisions

Consider the planar polynomial system \[ {dx\over dt}= H_t+\varepsilon P_5(x,y),\quad {dy\over dt}= -H_x+\varepsilon Q_5(x,y),\tag{\(*\)} \] where \(\varepsilon\) is a small parameter, \(\deg P_5= \deg Q_5= 5\), and \[ H(x,y):= x^2+ y^2-{3\over 4} (x^4+ y^4)+{1\over 6} (x^6+ y^6). \] Under the assumption that \((*)\) is \(\mathbb{Z}_4\)-equivariant, the authors prove that \((*)\) can have 28 limit cycles, the corresponding phase portraits are described.