In　this　paper,　the　radial　nonlinear　vibrations　are　investigated　for　a　thin-walled　hyperelastic　cylindrical　shell　composed　of　the　classical　incompressible　Mooney–Rivlin　materials　subjected　to　a　radial　harmonic　excitation.　Using　Lagrange　equation,　Donnell’s　nonlinear　shallow-shell　theory　and　small　strain　assumption,　the　nonlinear　differential　governing　equation　of　motion　is　obtained　for　the　incompressible　Mooney–Rivlin　material　thin-walled　hyperelastic　cylindrical　shell.　The　differential　governing　equation　of　motion　is　simplified　to　a　generalized　Duffing　equation　with　the　quadratic　term.　The　second-order　approximate　analytical　solutions　are　obtained　by　using　the　modified　Lindstedt–Poincaré　(MLP)　method.　The　impacts　of　the　parameters　on　the　amplitude–frequency　response　curves　and　number　of　the　equilibrium　points　are　analyzed.　According　to　Runge–Kutta　method,　the　bifurcation　diagrams,　Lyapunov　exponents　and　Poincaré　maps　are　obtained.　The　chaotic　behaviors　are　found　in　the　radial　nonlinear　vibrations　of　the　incompressible　Mooney–Rivlin　material　thin-walled　hyperelastic　cylindrical　shell.　The　results　demonstrate　that　the　nonlinear　dynamic　responses　of　the　incompressible　Mooney–Rivlin　material　thin-walled　hyperelastic　cylindrical　shell　are　highly　sensitive　to　the　structural　parameters　and　external　excitation.　©　2023,　The　Author(s),　under　exclusive　licence　to　Springer　Nature　B.V.