Fractional order logistic map; Numerical approach

Marius-F. Danca

incompletemedium confidence

Category: Not specified
Journal tier: Note/Short/Other
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper numerically observes (i) initial-condition–dependent bifurcation sets and (ii) very long transients that require thousands of iterations, but explicitly provides no analytical proof. The model supplies rigorous arguments: trivial non-uniqueness via the equilibria x≡0 and x≡1 for any scan, a stronger injectivity claim when the p-scan interval contains 0, and two mechanisms (small p; small q with Mittag–Leffler decay) yielding arbitrarily long transients with quantitative bounds. The only mismatch is scope: the paper asserts “for every initial condition another BD is obtained” across typical scans (often with p not including 0), whereas the model’s global non-uniqueness proof uses the equilibria or the p=0 endpoint; still, the core claims are upheld and clarified.

Referee report (LaTeX)

\textbf{Recommendation:} major revisions

\textbf{Journal Tier:} note/short/other

\textbf{Justification:}

The manuscript documents two noteworthy numerical phenomena for the Caputo-like discrete fractional logistic map: initial-condition dependence of bifurcation overlays and very long transients. The numerical evidence is convincing, the implementation details are transparent, and the phenomena are of interest to practitioners. However, the central claims lack theoretical support (as the authors acknowledge). The paper would be substantially strengthened by adding analytical lemmas (e.g., on equilibria, invariant regions, and fractional Gronwall-based decay) that at least partially explain the observations and by clarifying the precise meaning of the displayed bifurcation sets.