Fractional gradient system and generalized Birkhoff system

New fractional gradient representation of generalized Birkhoff system is studied. New definition of fractional gradient system for generalized Birkhoff system is given. New condition for generalized Birkhoff system transforms into arbitrary fractional-order (integer or non-integer) gradient system is deduced. The fractional dynamical modelling method by fractional gradient system is also discussed. We apply our results into several classical nonlinear models, such as van der Pol equation, Hénon–Heiles equation and four-order generalized Birkhoff’s equation.