In　this　paper,　the　Fuzzy　Generalized　Cell　Mapping　(FGCM)　method　is　developed　with　the　help　of　the　Adaptive　Interpolation　(AI)　in　the　space　of　fuzzy　parameters.　The　adaptive　interpolation　on　the　set-valued　fuzzy　parameter　is　introduced　in　computing　the　one-step　transition　membership　matrix　to　enhance　the　efficiency　of　the　FGCM.　For　each　of　initial　points　in　the　state　space,　a　coarse　database　is　constructed　at　first,　and　then　interpolation　nodes　are　inserted　into　the　database　iteratively　each　time　errors　are　examined　with　the　explicit　formula　of　interpolation　error　until　the　maximal　errors　are　just　under　the　error　bound.　With　such　an　adaptively　expanded　database　on　hand,　interpolating　calculations　assure　the　required　accuracy　with　maximum　efficiency　gains.　The　new　method　is　termed　as　Fuzzy　Generalized　Cell　Mapping　with　Adaptive　Interpolation　(FGCM　with　AI),　and　is　used　to　investigate　codimension-Two　bifurcations　in　two-dimensional　and　three-dimensional　nonlinear　dynamical　systems　with　fuzzy　noise.　It　is　found　that　global　changes　in　fuzzy　dynamics　are　dominated　by　the　underlying　deterministic　counterparts,　and　the　fuzzy　attractor　expands　along　the　unstable　manifold　leading　to　a　collision　with　a　saddle　when　a　bifurcation　occurs.　The　examples　show　that　the　FGCM　with　AI　has　a　thirtyfold　to　fiftyfold　efficiency　over　the　traditional　FGCM　to　achieve　the　same　analyzing　accuracy.　©　2019　World　Scientific　Publishing　Company.