Stability analysis of fixed point of fractional-order coupled map lattices

Both the paper and the model reduce the linear fractional-order lattice to scalar eigenmodes via the Z-transform, obtain the same characteristic equation det(z(1−z^{-1})^α I − (A−I))=0, and identify the cardioid boundary β(t) by substituting z=e^{it}. They then specialize to 1D circulant A, giving the same spectra and the same stability regions for the symmetric (polygon via real slice) and asymmetric (family of scaled cardioids γ_j) cases. A small nuance: the paper states a linear real-axis bound 1−2α<λ<1 (used to produce linear polygonal regions) even though the exact cardioid slice is [1−2^α,1]; the model explicitly notes it uses the narrower [1−2α,1] only as a sufficient condition. Overall, the arguments align and are essentially the same, with the model adding minor analytic details (branch choice and Nyquist/argument principle). Key steps and formulas match the paper’s Eq. (10), its β(t) parameterization, circulant-eigenvalue formulas, and Theorems 3.1–3.4 as stated in the PDF .