Perturbation to Noether symmetry for fractional dynamic systems of variable order

Abstract

Fractional dynamic system of variable order is one of the interesting topics to study. Perturbation to symmetry, which is closely related to its integrability, is worth to be well studied. In this paper, Perturbation to Noether symmetry and adiabatic invariant are investigated in terms of Riemann–Liouville fractional derivative of variable order for fractional generalized Birkhoffian system. Then under some special conditions and transformations, adiabatic invariants for the fractional Birkhoffian system, the fractional Hamiltonian system and the fractional Lagrangian system are discussed. It has been observed that some results obtained from special cases are new and others are consistent with the existing results, which reconfirm the credibility of proposed achievements. Finally, an example is given to illustrate the methods and results.