Limit cycles of a \(Z_3\)-equivariant near-Hamiltonian system
The authors study the number of limit cycles number for the following near-Hamiltonian polynomial system \[ \dot x= 3y+ 8xy+ x^2y+y^3+ \varepsilon p(x,y), \quad \dot y= -3x+ 4x^2- 4y^2- x^3- xy^2+ \varepsilon q(x,y), \] where \(\varepsilon \geq 0\) is a small real parameter, \(p\) and \(q\) are polynomials of degree 5, 4 or 3. By using the method of Melnikov function it is proved that such a system with degrees 3, 4 and 5 can has 4, 10 and 15 limit cycles, respectively. The corresponding distributions of limit cycles are given.
scientific article; zbMATH DE number 5577452