Memory-Dependent Shifts of the Period-Doubling Cascade in the Finite-Memory Grünwald–Letnikov Fractional Logistic Map

Fractional logistic maps introduce memory into a canonical route to chaos, but how memory shifts successive period-doubling bifurcation points is not yet fully understood. For the finite-memory Grünwald–Letnikov (GL) fractional logistic map, we formulate equations that determine exact periodic orbits and their period-doubling points, with the number of unknowns independent of memory length. The framework reproduces an independently derived exact fixed-point boundary and identifies a genuine period-4 to period-8 doubling, confirming its applicability beyond the fixed point. Across the first seven computed bifurcation levels, increasing memory length shifts the bifurcation points toward smaller control-parameter values, with stronger shifts at lower fractional orders. The GL memory tail decays more slowly at lower fractional order, helping explain why extending retained memory produces larger bifurcation-point shifts. At fixed memory length, decreasing the fractional order from its classical value first moves the bifurcation points toward smaller control-parameter values, before a turnover carries them toward larger values. As the order approaches zero, the GL memory terms vanish and limiting bifurcation points lie one unit above their classical counterparts, so branch continuity requires a turnover. Together, these results provide a validated basis for computing and interpreting how memory reorganizes the period-doubling route to chaos.