This paper investigates the dynamic behavior of a second-order nonlinear rational difference equation modeling a population system with nonlinear interactions between current and previous population states. We derive analytical conditions for the stability of fixed points, explore codim-1 bifurcations, and compute the associated topological normal forms. The analysis also establishes the existence of period-2 solutions and reveals the potential for chaotic dynamics within specific parameter ranges. To validate the theoretical findings, we conduct numerical simulations and bifurcation analysis using the MATLAB package MatContM (version 5p4). Chaotic behavior is further confirmed through the computation of the largest Lyapunov exponent. The results offer new insights into the complex dynamics of delayed population models with nonlinear feedback, extending classical models and suggesting potential applications in stochastic systems and epidemiological modeling.

In this paper, we introduce the inclusive subfamilies of complex order $E({\delta }_1,{\delta }_2,{\delta }_3,{\delta }_4,a,b)$ and $C({\delta }_1,{\delta }_2,{\delta }_3,{\delta }_4,a,b)$, defined by means of the Liouville–Caputo-type derivative operator and subordination to the Horadam polynomials. For these subfamilies, we derive estimates for the initial coefficients $|q_2|$ and $|q_3|$, as well as results concerning the Fekete–Szegö functional $|q_3-\varrho q_2^2|$. In addition, several related results are established as corollaries, accompanied by a concluding remark.

This paper investigates the global dynamics of a broad class of nonlinear rational difference equations given by $x_{n+1}={\displaystyle \frac{a{x}_{n+1-2k}}{b+c{x}_{n+1-k}{x}_{n+1-2k}}},n=0,1,\dots ,$ which generalizes several known models in the literature. We establish the existence of exactly three equilibrium points and show that the trivial equilibrium is globally asymptotically stable when the parameter ratio $\alpha =(b/a)$ lies in $(-1,1)$. The nontrivial equilibria are shown to be always unstable. An explicit general solution is derived, enabling a detailed analysis of solution behavior in terms of initial conditions and parameters. Furthermore, we identify and classify minimal period $2k$ and $4k$ solutions, providing necessary and sufficient conditions for the occurrence of constant and periodic behaviors. These analytical results are supported by numerical simulations, confirming the theoretical predictions. The findings generalize and refine existing results by offering a unified framework for analyzing a wide class of rational difference equations.

The present work centers on the significance of q-calculus in geometric function theory and its expanding applications within the domain of Te-univalent functions, especially those associated with special polynomials like the q-Bernoulli polynomials. Motivated by recent interest in these polynomials, our study introduces and analyzes a generalized subclass of Te-univalent functions that intimately relate to q-Bernoulli polynomials. For this new family, we establish explicit bounds for $|d_2|$ and $|d_3|$, and provide estimates for the Fekete–Szegö functional $|d_3-\xi d_2^2|, \xi \in \mathbb{R}$. Our findings contribute new results and demonstrate meaningful connections to prior work involving Te-univalent and subordinate functions, thereby broadening and integrating various strands of the existing literature.

This research paper introduces a method utilizing the Maclaurin series to analyze the standard Sakiadis problem. The technique uses the Maclaurin series to describe fluid mechanics boundary layers, combining the series solution with diagonal Padé approximants for handling the infinity condition. Additionally, the Hausdorff derivative is used to examine the Sakiadis equation's fractal formulation. The current approach is consistent with the previous process, and it follows the correct balance. This study is an important resource for furthering research in this area and provides insightful information.

Coefficient operators provide a natural way to reshape an analytic function before testing its geometry. This paper develops a two-parameter algebraically damped Hadamard-product operator and investigates the functions whose transformed images satisfy a disk-starlikeness condition. The construction retains a visible contribution from the first coefficients while allowing the higher-order tail to be attenuated at an adjustable algebraic rate. The geometric conclusion concerns the transformed function; it does not automatically imply that the original function is starlike or univalent. Sharp initial-coefficient estimates, a corrected Fekete–Szegö inequality, and corresponding extremal functions are obtained. The multiplier is compared quantitatively with several established coefficient operators, and the effects of its parameters on damping and coefficient flexibility are examined. A weighted-coefficient criterion yields transparent perturbation and convex-combination consequences, while explicit examples illustrate both membership and failure of the criterion. The practical profiles included in the paper are conceptual visualizations rather than empirically validated models.