Fractional-Order Logistic Model Yields Map-Based Route to Bifurcation and Stabilization

A new study in Boundary Value Problems analyzes a logistic population model that combines a piecewise constant delayed argument with an interval-reset conformable derivative, working out when its equilibrium loses stability and how feedback control can restore it. The paper, authored by Hasan Gündüz, Güven Kaya and Ercan Çelik, was published on 27 October 2026 as an open-access research article.

By integrating the model exactly on each sampling interval, the authors reduce the continuous system to a two-dimensional map, including a continuous extension at the map’s apparent singularity. A Jury stability analysis applied to this map yields an explicit threshold for the stability of the positive equilibrium. The team then evaluates the unit-circle crossing conditions, low-order nonresonance conditions, and the cubic normal-form coefficient needed to characterize a Neimark–Sacker bifurcation.

The study also examines three established control strategies—OGY-type parameter feedback, hybrid control, and exponential feedback—using local stability conditions, and runs a common stabilization experiment that compares controller gains, spectral radii, and settling iterations across the methods. To probe whether the dynamics become chaotic beyond the bifurcation, the authors recompute phase portraits and finite-time maximum Lyapunov exponents. Notably, for the reported growth-rate scan, the exponents beyond the bifurcation are predominantly close to zero, so the analysis does not establish a chaotic regime; the exponents instead distinguish loss of equilibrium stability from genuine evidence of chaos.

The authors are explicit about the scope of their contribution: it is a model-specific analytical and numerical comparison rather than a new fractional operator or a new control algorithm. They also caution that the conformable order acts through an effective interval length and should not be interpreted as a nonlocal memory mechanism. The article is being shared early as a peer-reviewed, accepted, citable version with a permanent DOI, subject to further edits before the final Version of Record.

Subject of Research: Bifurcation analysis and feedback stabilization of a conformable fractional-order logistic population model with delay

Article Title: Bifurcation and chaos control in a conformable fractional-order logistic system

Article References: Gündüz, H., Kaya, G., & Çelik, E. (2026). Bifurcation and chaos control in a conformable fractional-order logistic system. Boundary Value Problems. https://doi.org/10.1186/s13661-026-02378-7

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DOI: 10.1186/s13661-026-02378-7

Keywords: Neimark–Sacker bifurcation, conformable fractional order, logistic model, delayed argument, Jury stability, chaos control, OGY feedback, hybrid control, exponential feedback, Lyapunov exponents, two-dimensional map, stability threshold