Bifurcation Analysis and Basin Fragmentation in a Population Model

In this paper, we investigate the dynamics of a two-dimensional Hénon-like map obtained by adding a state-dependent linear feedback control to a variant of the logistic map that explicitly incorporates carrying capacity. We establish the local stability regions in the parameter plane and identify codimension-1 (transcritical, flip, and Neimark–Sacker) bifurcation boundaries. We prove the degeneracy of the Neimark–Sacker bifurcation along a parameter curve. We construct normal form maps for the [Formula: see text], [Formula: see text], and [Formula: see text] strong resonances, demonstrating that only the [Formula: see text] resonance is nondegenerate. We highlight persistent chaos via period-doubling cascades and transient chaos driven by resonant heteroclinic tangles. Finally, by interpreting the system as a population model subject to an external intervention, we illustrate how control intensity induces severe fragmentation of basins of attraction.