Feigenbaum constant universality in fractional and fractional difference maps

Prove that the Feigenbaum constant δ describing the scaling of successive parameter intervals at period-doubling bifurcations exists for fractional and fractional difference maps and that its value equals the classical Feigenbaum constant observed in regular (integer-order) maps, i.e., establish that the ratios of consecutive parameter spacings between period-doubling bifurcation points converge to δ ≈ 4.6692016 in fractional logistic and related fractional maps.

Background

The paper computes asymptotic period-doubling bifurcation points for the fractional logistic map and the fractional difference logistic map (e.g., at order α = 0.5) and observes that the ratios of successive parameter intervals oscillate around the classical Feigenbaum number, with apparent convergence albeit slower than in regular maps due to power-law convergence inherent in fractional dynamics.

Based on these numerical results and analytical formulations for bifurcation points, the authors explicitly advance a conjecture that the Feigenbaum constant exists in fractional and fractional difference maps and has the same value as in regular maps, calling for a formal proof of this universality.