We explore the local dynamical characteristics, chaos and bifurcations of a two-dimensional discrete laser model in ${\mathbb{R}}_{+}^2$. It is shown that for all $a$, $b$, $c$ and $p$, model has boundary fixed point $P_{0y}(0,\frac{p}{c})$, and the unique positive fixed point $P_{xy}^{+}(\frac{ap-bc}{ab},\frac{b}{a})$ if $p>\frac{bc}{a}$. Further, local dynamical characteristics with topological classifications for the fixed points $P_{0y}(0,\frac{p}{c})$ and $P_{xy}^{+}(\frac{ap-bc}{ab},\frac{b}{a})$ have explored by stability theory. It is investigated that flip bifurcation exists for the boundary fixed point $P_{0y}(0,\frac{p}{c})$, and also there exists a flip bifurcation if parameters vary in a small neighborhood of the unique positive fixed point $P_{xy}^{+}(\frac{ap-bc}{ab},\frac{b}{a})$. Moreover, it is also explored that for the fixed point $P_{xy}^{+}(\frac{ap-bc}{ab},\frac{b}{a})$, laser model undergoes a Neimark-Sacker bifurcation, and in the meantime stable invariant curve appears. Numerical simulations are implemented to verify not only obtain results but also exhibit complex dynamics of period $-2$, $-3$, $-4$, $-5$, $-8$ and $-9$. Further, maximum lyapunov exponents along with fractal dimension are computed numerically to validate chaotic behavior of the laser model. Lastly, feedback control method is utilized to stabilize chaos exists in the model.

In this paper, we explore local dynamic characteristics, bifurcations and control in the discrete activator-inhibitor system. More specifically, it is proved that discrete-time activator-inhibitor system has an interior equilibrium solution. Then, by using linear stability theory, local dynamics with different topological classifications for the interior equilibrium solution are investigated. It is investigated that for the interior equilibrium solution, discrete activator-inhibitor system undergoes Neimark-Sacker and flip bifurcations. Further chaos control is studied by the feedback control method. Finally, numerical simulations are presented to validate the obtained theoretical results.

We explored a local stability analysis at fixed points, bifurcations, and a control in a discrete Leslie's prey-predator model in the interior of ${\mathbb{R}}_{+}^2$. More specially, it is examined that for all parameters, Leslie's model has boundary and interior equilibria, and the local stability is studied by the linear stability theory at equilibrium. Additionally, the model does not undergo a flip bifurcation at the boundary fixed point, though a Neimark-Sacker bifurcation exists at the interior fixed point, and no other bifurcation exists at this point. Furthermore, the Neimark-Sacker bifurcation is controlled by a hybrid control strategy. Finally, numerical simulations that validate the obtained results are given.

In this paper, we investigated the local stability, chaotic dynamics, and bifurcations of a discrete tuberculosis (TB) epidemic model, which provides crucial insights into the complex behavior of infectious disease spread in discrete-time settings. Specifically, we established the existence of a disease-free fixed point for all parametric values and demonstrated that an endemic fixed point emerges under specific parametric condition. The local stability at these fixed points was examined using the linear stability theory, revealing important thresholds for disease persistence or eradication. A comprehensive bifurcation analysis was conducted, showing that while no flip bifurcation occurred at the disease-free fixed point, both Neimark-Sacker and flip bifurcations took place at the endemic fixed point, and we studied said bifurcations by the explicit criterions. To further understand the model's nonlinear dynamics, we explored chaotic behavior by a feedback control strategy to mitigate chaotic oscillations. Numerical simulations are presented to validate our theoretical findings, demonstrating the model's capacity to capture real-world epidemiological patterns. Finally, theoretical results are also interpreted biologically/medically.

Mathematical models play a crucial role in understanding the dynamics of epidemic diseases by providing insights into how they spread and be controlled. In biomathematics, mathematical modeling is a powerful tool for interpreting the experimental results of biological phenomena related to disease transmission, offering precise and quantitative insights into the processes involved. This paper focused on a discrete mathematical model of the Hepatitis C virus (HCV) to analyze its dynamical behavior. Initially, we examined the local dynamics at steady states, providing a foundation for understanding the system's stability under various conditions. We then conducted a detailed bifurcation analysis, revealing that the discrete HCV model undergoes a Neimark-Sacker bifurcation at the uninfected steady state. Notably, our analysis showed that no period-doubling or fold bifurcations occur at this state. Further investigation at the infected steady state demonstrated the presence of both period-doubling and Neimark-Sacker bifurcations, which are characterized using explicit criteria. By employing a feedback control strategy, we explored chaotic behavior within the HCV model, highlighting the complex dynamics that can arise under certain conditions. Numerical simulations were conducted to verify the theoretical results, illustrating the model's validity and applicability. From a biological perspective, the insights gained from this analysis enhance our understanding of HCV transmission dynamics and potential intervention strategies. The presence of Neimark-Sacker bifurcation at the uninfected steady state implies that small perturbations could lead to oscillatory behavior, which may correspond to fluctuations in the number of infections over time. This finding suggests that maintaining stability at this steady state is critical for preventing outbreaks. The period-doubling and Neimark-Sacker bifurcations at the infected steady state indicate the potential for more complex oscillatory patterns, which could represent persistent cycles of infection and remission in a population. Finally, exploration of chaotic dynamics through feedback control highlights the challenges in predicting disease spread and the need for careful management strategies to avoid chaotic outbreaks.

In this paper, we explore the bifurcation, chaos, and local stability of a discrete Hepatitis C virus infection model. More precisely, we studied the local stability at fixed points of a discrete Hepatitis C virus model. We proved that at a partial infection fixed point, the discrete HCV model undergoes Neimark-Sacker bifurcation, but no other local bifurcation exists at this fixed point. Moreover, it was also proved that period-doubling bifurcation does not occur at liver-free, disease-free, and total infection fixed points. Furthermore, we also examined chaos control in the understudied discrete HCV model. Finally, obtained theoretical results were confirmed numerically.