Dynamics of a 3D-Noninvertible Map via Lambert-Type Functions: Bifurcations, Arnold Tongues, Critical Manifolds and Preimages

This study investigates a Three-Dimensional (3D)-Ricker endomorphism obtained by embedding the classical one-dimensional Ricker population model into a smooth, noninvertible discrete-time system. The equilibrium structure is analytically characterized through a Lambert [Formula: see text] function. Explicit expressions for the fixed points are derived, yielding conditions for their existence and multiplicity, and enabling the identification of parameter regions with one, two, or three coexisting equilibria. The local dynamics are investigated through a bifurcation analysis of both trivial and nontrivial fixed points. In particular, the fold–Neimark–Sacker and flip–Neimark–Sacker bifurcations are characterized at the trivial fixed point, as well as the fold and flip bifurcations at the nontrivial fixed points. Furthermore, it is shown that the fold bifurcation surface is intrinsically connected to the branching condition of the Lambert [Formula: see text] function. The dynamics arising from the Neimark–Sacker bifurcation are further examined through the computation of Arnold isoclines and Arnold tongues. These resonance regions indicate frequency-locking phenomena and describe the transition from invariant closed curves to periodic and chaotic attractors. The noninvertible nature of the 3D-Ricker endomorphism is analyzed via its critical manifolds and preimage structure. Analytical expressions for the critical manifolds are derived using Lambert-type transcendental functions, while a [Formula: see text]-Lambert function yields exact conditions for determining the number of preimages of arbitrary points. The relationship between critical manifolds and preimage distributions is investigated through representative phase-space cross-sections. Overall, the results provide a unified description of the main dynamical and geometrical properties of the proposed 3D-Ricker endomorphism.