Libration　points　are　the　five　equilibrium　solutions　in　the　circular　restricted　three-body　problem　(CRTBP).　The　linearized　motions　around　triangular　libration　points　are　typical　center　×　center　type.　Studies　about　probes　moving　around　orbits　in　the　vicinity　of　the　libration　points　have　theoretical　significance.　From　the　vibrational　point　of　view,　the　polynomial　series　are　used　to　derive　approximately　the　relations　in　different　directions　during　periodic　motions,　which　provides　a　new　point　of　view　to　exploring　the　dynamics　and　analyzing　the　overall　characteristics　of　the　whole　system　with　general　rules.　The　nonlinear　relations　in　polynomial　form　between　the　directions　of　the　planar　motions　can　be　treated　as　constraints　to　obtain　the　solutions　by　numerical　integration.　Numerical　simulations　verify　the　efficiency　of　the　proposed　method.　The　methodology　of　deriving　topological　relations　has　the　potential　to　be　extended　to　circular/elliptical　R3BP　in　three　dimensional　cases.　©　2017,　Editorial　Office　of　Chinese　Journal　of　Theoretical　and　Applied　Mechanics.　All　right　reserved.