Bifurcation Sets and Distributions of Limit Cycles in a Hamiltonian System Approaching the Principal Deformation of a Z4-FIELD

We consider a perturbed Hamiltonian system approaching the principal deformation of a Z4-equivariant principal singular field. By studying global and local bifurcations of this system, nine limit cycles with the distribution $C_9^1 \supset 4C_1^2 $ and five limit cycles with the II′jasenko distribution $5C_1^1 $ are found. Explicit formulas for the bifurcation sets and their phase portraits have also been obtained.