Audounet–Matignon–Montseny conjecture on scalar nonlinear fractional stability

Consider the scalar nonlinear Caputo fractional differential system of order $0<\alpha<1$

$$^{C\!}D_{0+}^{\alpha}x(t)=f(x(t)),$$

with equilibrium $x^{*}=0$, where $f'(0)=\lambda\in\mathbb{C}$. Its linearization at the equilibrium is

$$^{C\!}D_{0+}^{\alpha}x(t)=\lambda x(t).$$

Audounet–Matignon–Montseny conjecture. The local stability of $x^{*}=0$ is governed by the global stability of the linearized system as follows: (i) $x^{*}=0$ is locally asymptotically stable if $|\arg(\lambda)|>\frac{\alpha\pi}{2}$; (ii) $x^{*}=0$ is not locally stable if $|\arg(\lambda)|<\frac{\alpha\pi}{2}$.

The conjecture concerns whether the stability or instability of the scalar linearized fractional system determines the corresponding local behavior of the nonlinear system. The source presents the instability direction as an open question, while noting that the asymptotically stable direction had been established for the broader system considered there.