This　paper　studies　the　periodic　solutions　of　a　four-dimensional　coupled　polynomial　system　with　N-degree　homogeneous　nonlinearities　of　which　the　unperturbed　linear　system　has　a　center　singular　point　in　generalization　resonance　1　:　n　at　the　origin.　Considering　arbitrary　positive　integers　n　and　N　with　n　＜=　N　and　N　＞=　2,　the　new　explicit　expression　of　displacement　function　for　the　four-dimensional　system　is　detected　by　introducing　the　technique　on　power　trigonometric　integrals.　Then　some　precise　and　detailed　results　in　comparison　with　the　existing　works,　including　the　existence　condition,　the　exact　number,　and　the　parameter　control　conditions　of　periodic　solutions,　are　obtained,　which　can　provide　a　new　theoretical　description　and　mechanism　explanation　for　the　phenomena　of　emergence　and　disappearance　of　periodic　solutions.　Results　obtained　in　this　paper　improve　certain　existing　results　under　some　parameter　conditions　and　can　be　extensively　used　in　engineering　applications.　To　verify　the　applicability　and　availability　of　the　new　theoretical　results,　as　an　application,　the　periodic　solutions　of　a　circular　mesh　antenna　model　are　obtained　by　theoretical　method　and　numerical　simulations.