Equivalent system for a multiple-rational-order fractional differential system

The equivalent system for a multiple-rational-order (MRO) fractional differential system is studied, where the fractional derivative is in the sense of Caputo or Riemann–Liouville. With the relationship between the Caputo derivative and the generalized fractional derivative, we can change the MRO fractional differential system with a Caputo derivative into a higher-dimensional system with the same Caputo derivative order lying in (0,1). The stability of the zero solution to the original system is studied through the analysis of its equivalent system. For the Riemann–Liouville case, we transform the MRO fractional differential system into a new one with the same order lying in (0,1), where the properties of the Riemann–Liouville derivative operator and the fractional integral operator are used. The corresponding stability is also studied. Finally, several numerical examples are provided to illustrate the derived results.