Stability properties of two-term fractional differential equations

This paper formulates explicit necessary and sufficient conditions for the local asymptotic stability of equilibrium points of the fractional differential equation $$\begin{aligned} {D}^{\alpha }y(t)+f(y(t),\, {D}^{\beta }y(t))=0,\quad t>0 \end{aligned}$$Dαy(t)+f(y(t),Dβy(t))=0,t>0involving two Caputo derivatives of real orders $$\alpha >\beta $$α>β such that $$\alpha /\beta $$α/β is a rational number. First, we consider this equation in the linearized form and derive optimal stability conditions in terms of its coefficients and orders $$\alpha ,\beta $$α,β. As a byproduct, a special fractional version of the Routh–Hurwitz criterion is established. Then, using the recent developments on linearization methods in fractional dynamical systems, we extend these results to the original nonlinear equation. Some illustrating examples, involving significant linear and nonlinear fractional differential equations, support these results.