Threshold-equality conjecture for commensurate fractional-order chaotic systems

Consider a commensurate fractional-order system

$${}_0^C\mathrm{D}_t^{\alpha}X=f(X),$$

with threshold value $\alpha_t$, and let $\bar{\alpha}$ be the maximum stability bound $\alpha_*(E)$ over the equilibrium points associated with the system's chaotic attractor. Threshold-equality conjecture. If the system is chaotic with threshold value $\alpha_t$, and $\bar{\alpha}$ corresponds to the equilibrium points associated with the chaotic attractor, then

$$\alpha_t=\bar{\alpha}.$$

The conjecture proposes that the numerically observed onset threshold for chaos equals the largest equilibrium-point stability bound. In the example discussed, the authors observe $\alpha_t\geq\bar{\alpha}$ and find both values to be $0.940887$; the general equality is presented as a conjecture based on numerical observations.