| This work is involved in studying the second part of Hilbert's 16th problem which is related to finding the maximal number and relative locations of limit cycles of planar polynomial vector fields for given degree n.;Further, a systematic procedure has been explored to study general Zq-equivariant planar polynomial Hamiltonian vector fields for the maximal number of closed orbits and the maximal number of limit cycles after perturbation. This procedure is still under its early stage of development. Following the procedure by taking special consideration of Z 12 symmetric vector fields of degree 11, a maximum 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-degree systems as: H(11) ≥ 112. Two conjectures are proposed regarding the maximal number of closed orbits for equivariant polynomial Hamiltonian vector fields and the raised lower bound of the number of limit cycles bifurcated from the well defined Hamiltonian vector fields after perturbation.;This research is also extended to study bifurcations of limit cycles on rarely considered even degree polynomial vector fields. A sixth-degree polynomial is added to a fifth-degree symmetric polynomial Hamiltonian system. To obtain the maximal possible number of limit cycles, both local and global bifurcations are considered. By employing the detection function method for global bifurcations and normal form theory for local degenerate Hopf bifurcations, 31 and 35 limit cycles and their locations are obtained from two different sets of controlled parameters.;At first, Z10-equivariant polynomial Hamiltonian vector fields of degree 9 are studied to find the maximal possible number of closed orbits around the system's critical points. Optimized by coefficient control techniques, 63 closed orbits are realized from a ninth-degree Z10-equivariant Hamiltonian vector field. And further, by controlling perturbation parameters using the detection function method, a total of 80 limit cycles bifurcated from the perturbed Hamiltonian vector field with 3 different configurations, which gives rise to H(9) ≥ 92 - 1. |
