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

In this article, a systematic procedure has been explored to studying general Z(q)-equivariant planar polynomial Hamiltonian vector fields for the maximal number of closed orbits and the maximal number of limit cycles after perturbation. Following the procedure by taking special consideration of Z(12)-equivariant vector fields of degree 1.1, the maximal of 99 closed orbits are obtained under a well-defined coefficient group. Consequently, perturbation parameter control in limit cycle computation leads to the existence of 121 limit cycles in the perturbed Hamiltonian vector field, which gives rise to the lower bound of Hilbert number of 11th-order systems as H(11) >= 11(2). Two conjectures are proposed regarding the maximal number of closed orbits for equivariant polynomial Hamiltonian vector fields and the maximal number of limit cycles bifurcated from the well defined Hamiltonian vector fields after perturbation. (c) 2005 Elsevier Ltd. All rights reserved.