Existence of 121 limit cycles in a perturbed planar polynomial Hamiltonian vector field of degree 11 (Q939458)

The paper explores a systematic route to find the maximal number of limit cycles of perturbed planar polynomial \(Z_q\)-equivariant Hamiltonian system. Thus, the paper is devoted to the investigation of the weakened 16-th Hilbert's problem proposed by Arnold. The first step in searching limit cycles is to define an optimized \(Z_q\)-equivariant Hamiltonian polynomial system with the most possible number and type of closed orbits, compound cycles and oval schemes. The equations determining the coefficients of the polynomial Hamiltonian function for the desired level curve structure have been constructed. Taking special consideration of \(Z_{12}\)-equivariant vector fields of degree \(11\), \(99\) closed orbits are obtained. Computing the global bifurcation of limit cycles gives rise to the maximum of \(121\) limit cycles. Two conjecture are proposed regarding the maximal number of closed orbits for equivariant Hamiltonian polynomial vector fields and the maximal number of limit cycles bifurcating from well defined Hamiltonian vector fields.