Noether’s theorem for fractional Birkhoffian system of Herglotz type with time delay

•The Pfaff-Herglotz principle of fractional Birkhoffian syetem with time delay was established.•The fractional Birkhoff’s equation with time delay based on Herglotz generalized variational principle was derived.•Noether’s theorem of fractional Birkhoffian systems with time delay based on Herglotz generalized variational principle was proposed and proved. Noether symmetry theorem of Herglotz type for time-delayed fractional Birkhoffian system are studied. Firstly, based on the fractional derivative of Riemann-Liouville, the Herglotz variational principle of time-delayed fractional Birkhoffian system is established, and the time-delayed Birkhoff′s equation of Herglotz type is derived. Secondly, the definition and criterion of Herglotz type Noether symmetric transformation of time-delayed fractional Birkhoffian system are established. Thirdly, the Noether′s theorem of the system is proposed and proved, in addition, the inner relationship between Noether symmetries and conservation is accurately explored. Next, the special case of the theorem is discussed, in other words, when the Herglotz generalized variational principle is reduced to the classical variational principle, the result of this paper is degraded into the Noether symmetry theorem of the time-delayed fractional Birkhoffian system. Finally, an example is given.