The　mathematical　model　of　vortex-induced　vibrations　(VIV)　on　long-span　bridges　is　important　to　predict　nonlinear　structural　responses.　Such　models　can　be　divided　into　two　categories:　wake-oscillator　and　single-degree-of-freedom　(SDOF)　models.　The　SDOF　model　is　widely　used　for　wind-induced　vibration　calculations.　However,　the　traditional　SDOF　model　based　on　the　standard　van　der　Pol　oscillator　cannot　simulate　VIVs　with　multistability.　In　this　study,　a　newly　generalized　van　der　Pol　model　is　proposed　to　incorporate　the　limit-cycle　oscillation　(LCO)　with　multiple　amplitudes,　and　the　nonlinear　damping　is　expressed　by　polynomial　expansion.　Next,　the　multiple　LCO　amplitudes　can　be　determined　from　the　energy　evolution　formula　derived　from　the　averaging　method.　Similarly,　the　evolution　of　the　vibration　amplitude　during　the　transient　response　is　also　derived　by　the　same　method.　Subsequently,　nonlinear　parameter　identification　methods　based　on　constraint　optimization　are　derived　according　to　both　the　LCO　amplitude　and　transient　responses.　In　the　last　part　of　this　study,　the　“energy　map”　is　proposed　to　present　the　energy　extracted　from　the　fluid–structure　interaction　with　different　wind　speeds　and　vibration　amplitudes,　and　it　is　constructed　by　the　parameters　identified　in　the　lock-in　range　of　VIV.　The　“energy　map”　can　provide　a　complete　picture　of　the　evolution　of　the　energy　of　VIVs　on　bridge　decks.　©　2023,　The　Author(s),　under　exclusive　licence　to　Springer　Nature　B.V.