Chaotic features in fractional iterative maps

We investigate several properties of the fractional iterative map with two control parameters of β and γ in the range of the fractional value 1⩽η⩽2. The relation between the control parameter γ∞ at an infinite sequence of period-doubling bifurcations and the fractional value η is shown for fixed values of β. The value of γ∞ is expected to be near four as η decreases from 2 to 1 for β<1.0. Especially, the numerical values of γ∞ are found to show approximately linear decay with decreasing η.