This　paper　focuses　on　the　investigation　of　the　dynamics　of　novel　2-DOF　coupled　oscillators.　The　system　consists　of　a　linear　oscillator　(main　structure)　and　an　attached　lightweight　nonlinear　oscillator,　called　a　nonlinear　energy　sink　(NES),　under　harmonic　forcing　in　the　regime　of　1:1:1　resonance.　The　studied　NES　has　geometrically　nonlinear　stiffness　and　damping.　Due　to　the　degeneracies　that　the　NES　brings　to　the　system,　diverse　bifurcation　structures　and　rich　dynamical　phenomena　such　as　nonlinear　beating　and　strongly　modulated　response　occur.　The　latter　two　phenomena　represent　different　patterns　of　energy　transfer.　To　capture　the　bifurcation　structure,　the　slow　flow　of　the　system　can　be　acquired　with　the　use　of　the　complex-averaging　method.　Furthermore,　by　applying　the　bifurcation　analysis　technique,　we　get　curve　boundaries　of　several　bifurcation　points　in　the　parameter　space.　These　boundaries　will　induce　different　types　of　folding　structures,　which　can　lead　to　complicated　patterns　of　strongly　modulated　responses,　in　which　intense　energy　transfer　from　the　main　structure　to　NES　occurs.　To　study　the　necessary　parameter　conditions　of　strongly　modulated　responses,　we　analyzed　the　dynamics　of　different　time　scales　of　the　slow　flow　in　detail　and　determined　the　corresponding　parameter　ranges　finally.　It　is　worth　noting　that　the　small　parameter　ε　may　have　a　qualitative　impact　on　the　dynamics　of　the　system.　©　2023,　The　Author(s),　under　exclusive　licence　to　Springer　Nature　B.V.