Computational Insights into the Unstable Fixed Point of the Fractional Difference Logistic Map

Ernestas Uzdila, Inga Telksniene, Tadas Telksnys and Minvydas Ragulskis ()
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Ernestas Uzdila: Department of Mathematical Modelling, Kaunas University of Technology, Studentu 50-147, LT-51368 Kaunas, Lithuania
Inga Telksniene: Department of Mathematical Modelling, Kaunas University of Technology, Studentu 50-147, LT-51368 Kaunas, Lithuania
Tadas Telksnys: Department of Mathematical Modelling, Kaunas University of Technology, Studentu 50-147, LT-51368 Kaunas, Lithuania
Minvydas Ragulskis: Department of Mathematical Modelling, Kaunas University of Technology, Studentu 50-147, LT-51368 Kaunas, Lithuania

Mathematics, 2024, vol. 12, issue 23, 1-13

Abstract: Thedivergence from the unstable fixed point of the fractional difference logistic map is investigated in this paper. In contrary to the classical logistic map, the memory horizon of the fractional difference logistic map reaches the initial condition. And though higher order orbits do not exist in the fractional difference logistic map, a trajectory started at the unstable fixed point may continuously remain at the fixed point as the number of iterations tends to infinity. Such an effect is well known for the classical logistic map, but less so in the fractional difference logistic map. It appears that this effect depends on the accuracy of the floating point arithmetic. It is demonstrated that the divergence from the unstable fixed point of the fractional difference logistic map is a completely computational artifact. Using double precision, approximately 32% values of a from the interval 2.7 < a ≤ 3.7 diverge from the unstable fixed point.

Keywords: periodic orbit; stability; fractional derivative; logistic map (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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