Asymptotic-autonomy principle for Caputo fractional systems

Establish an asymptotic-autonomy, or limit-system, principle for Caputo fractional systems that rigorously proves convergence of the awareness layer to its nontrivial rumor-free equilibrium distribution after rumor extinction when the awareness layer is supercritical.

Background

The fractional-order SIS-UAU model couples rumor transmission on a risk layer with awareness diffusion on an information layer. The paper proves global convergence to the fully unaware, rumor-free equilibrium under the sufficient conditions $\bar{\mathcal R}_0<1$ and $\lambda\rho(\mathbf A)<\delta$.

When the awareness layer is supercritical, meaning $\lambda\rho(\mathbf A)>\delta$, rumor extinction should leave a nontrivial awareness distribution at the rumor-free equilibrium. The authors explain that proving convergence to this distribution requires an asymptotic-autonomy or limit-system argument adapted to Caputo fractional dynamics. Such an argument is not supplied because Caputo flows lack the semigroup property used in the corresponding integer-order proof; the paper verifies the convergence only numerically.