Vol. 14, No. 2 (2025): Regular Issue

The work carried out an analytical and numerical analysis of the transition to space-time chaos in the nonlinear Kawahara equation through cascades of bifurcations of traveling waves in accordance with the universal bifurcation scenario of Feigenbaum-Sharkovsky-Magnitskii. It has been proven that the bifurcation parameter in this case is the propagation velocity of traveling waves along the spatial axis, which is clearly not included in the original equation.

The two dimensional unstable manifolds of delay neutrophil model are drawn as system loss its stability. The attractors are stable solutions and the unstable manifolds which are originated from the equilibrium solution form the neighborhood boundary of the related attractor. Under the parameter perturbation with periodical excitation, the two dimensional manifolds are also drawn since the system has stable attractor too. The two dimensional manifold is also obtained by periodical excitation via perturbation about its apoptosis rate. The observed phenomena usually illustrate the oscillation with multi-rhythm periodical solutions in neutrophil system.

In this article, we study the Semigeostrophic Equations in meteorology. These equations were introduced by Hoskins and Bretherton [1]. After suitable changes of variables, one can obtain the following coupled the Monge-Ampère/Transport problem $$\begin{aligned} \frac{\partial q}{\partial t} + J(\psi, q)=0 \\ \Delta \phi + \det(\frac{\partial^2 \phi}{\partial x_i \partial x_j})+ 1 = q\nonumber\\ \psi_{xx} \phi_{yy} - 2 \psi_{xy} \phi_{xy} + \psi_{yy} \phi_{xx} + \Delta \psi -\Delta \phi = 0 \end{aligned}$$ We proved the existence and uniqueness of time-periodic solution to the system.

This study investigates the outbreak of heroin addiction using a new model for heroin transmission with delay and a finite treatment period. The investigated model considers two different repulsion mechanisms, namely, repulsion from treatment to heroin addiction during treatment, and repulsion from treatment to potentially addicted individuals after treatment. These two repulsions make investigating the suggested age-structured model challenging. Indeed, we establish that the drug-free (resp. drug spread) equilibrium is locally asymptotically stable if the incidence rate $\beta$ is less than a certain threshold value $\beta^*$, with additional assumptions on the parameters of the model. Moreover, we have demonstrated the global asymptotic stability of the drug-free equilibrium when $\beta$ is less than another threshold value $\beta^{}$ (where $\beta^{**} < \beta^*$). Some numerical investigations of the model are conducted to identify effective measures for containing the epidemic.

In a two-dimensional vertical space, the equations determining the paths of a vehicle for which the magnitude of normal reaction force being constant are derived. Two different equations are considered: 1) Constant speed with energy not conserved, 2) Variable speed with energy conserved. The equations are cast in a non-dimensional form for universality of the results. Perturbation solutions, perturbation iteration method solutions (PIM) are derived for each case. In the case of vanishing normal force, instead of the approximate analytical solutions, the exact solutions are given. The approximate analytical solutions are contrasted with the numerical solutions. The critical value of the magnitude of the normal force which transforms the curves from concave-down to concave-up form is derived. It is found that the perturbation iteration solutions conform better to the numerical solutions than the perturbation solutions.

Camassa-Holm equation arises as model for unidirectional propagation of shallow water waves over a flat bottom. In this paper, we prove the well-posedness of the classical solution for the Cauchy problem associated with this equation, for every choice of the time $T$.

Over the last few decades, there has been a growing interest in chaotic systems and their behavior across research communities. Chaotic systems are dependent on initial conditions, meaning that even small variations can lead to vastly different outcomes, making precise prediction difficult. Synchronizing chaotic systems has become an important challenge with a wide range of applications, such as secure communication, robotics, and economics. However, synchronizing non-identical or differently ordered systems presents challenges. In common scenarios, synchronization algorithms and control strategies are derived under ideal conditions, but real-world applications are often affected by experimental uncertainties, external disturbances, and time delays, making control solutions more difficult to implement. As researchers explore the boundaries of this area of study, they have established several observations, including the challenges involved in synchronizing different types of chaotic systems and the importance of developing ways to address measured uncertainties and external disturbances. Overall, the study of chaotic system synchronization offers insights into a natural phenomenon that has broad applications across many disciplines.

The aim of this work is to investigate the existence and uniqueness of $S$-asymptotically Bloch type periodic solutions for a class of the neutral $\psi $-Hilfer fractional derivative equations with infinite delay. Our approach is based on the semigroup theory. In the end, we present an example to illustrate the applications of the abstract results.

The present investigation delves into the intricate dynamics of diabetic population, accounting for genetic, hereditary, social, environmental, and lifestyle determinants in the progression from pre-diabetes to diabetes. The model encompasses comorbidities, articulated through a suite of six nonlinear differential equations. Employing numerical methodologies alongside comprehensive stability and sensitivity analyses, it unveils nuanced insights into both biological and social interactions. Theoretical discoveries are vividly illustrated, and the model's credibility is attested through empirical validation. Conclusions drawn from the findings underscore pivotal parameters, endowing invaluable perspectives on the dynamical system in concert with stability elucidations.

In this study, we explore an ecological-epidemiological model involving phytoplankton and zooplankton with simultaneous harvesting of all species and a consideration of time delay. We incorporate the Holling type IV functional response to represent interactions between susceptible and infected phytoplankton, while phytoplankton predation is modeled using the Holling type I functional response. A unified harvesting effort (E) is applied to all species. We establish the positivity and boundedness of the solution, conduct feasibility and stability analyses for potential steady states, and investigate the existence of a bionomic equilibrium and an optimal harvesting policy using Pontryagin's maximal principle. Our findings indicate the stability of the trivial steady state when E > BTP (BTP=Biotechnical productivity of phytoplankton), with other states becoming asymptotically stable under specific conditions. A Hopf bifurcation analysis is conducted using harvesting and time delay as bifurcation parameters. Notably, both harvesting and time delay are highly sensitive to system dynamics, capable of inducing chaos. Elevating control parameters, such as the harvesting coefficient and the recovery rate of infected phytoplankton, as well as the growth rate of zooplankton derived from the predation of susceptible phytoplankton, plays a crucial role in stabilizing the system and mitigating chaos. Numerical simulations visually illustrate our theoretical results.

Understanding the environmental conditions associated with predators is essential to predator management. A tri-tropic food chain model is analyzed in this paper. The existence of a solution has been analyzed and shown to be uniformly bounded. The threshold number ($R_0$) is obtained, and the occurrence of bifurcation at $R_0=0$ is shown to be possible using central manifold theory. We used the Partial Rank Correlation Coefficient (PRCC) to do a global sensitivity analysis and identify the most sensitive parameters affecting $R_0$, providing information on potential ways to maintain ecological balance. Global stability of non-trivial equilibrium is established. Criteria for diffusion-driven ecological instability caused by local random movements of species are obtained. Detailed analyses of Turing patterns formation selected by the reaction-diffusion system under zero flux boundary conditions are presented. We found that $b_1$ and self-diffusion coefficients have an appreciable influence on the spatial spread of epidemics. Numerical simulation results confirm the analytical finding and generate patterns that indicate that the population and thus ecological balance can be maintained.

A deterministic compartment model for the transmission dynamics of coronavirus is formulated in this article. The impact of vaccine efficacy, media effect, and the cure rate of hospitalized individuals is investigated. The threshold quantity basic reproduction number ($R_0$) characterizes the SARS-CoV-2 transmission. It has been observed that the disease-free equilibrium is globally asymptotically stable for ${R}_0\le 1$, and the endemic equilibrium point is globally asymptotically stable for $R_0>1$. The global stability of both the equilibrium points is demonstrated via the Lyapunov function. The developed model has been fitted to the reported cumulative cases for the four most affected East Asian countries namely China, Japan, South Korea, and Taiwan to estimate the parameters. The numerical results indicate that increasing vaccine efficacy $ u_1$, treatment, and media effect can aid in controlling the spread of disease, reducing disease-induced mortality, and improving recovery.

The scaling parameters distinguish the fractal interpolation functions from the classical interpolation techniques. While generating hidden variable $A$-fractal function as an attractor for a specific iterated function system, the scaling parameters are considered in the form of upper triangular matrix. As the scalings can be chosen either as constants or functions, in this paper, for both the choices, the Weyl-Marchaud fractional derivative of $A$-fractal function is investigated. The essential conditions are enforced on the upper triangular matrix to demonstrate that the fractional derivative of $A$-fractal function is an attractor for a new iterated function system. To visualize the applications of fractal interpolation functions in the reconstruction process, the Lorenz and Rössler attractors of chaotic dynamical systems are reconstructed. Further, the generated fractal attractors consist of more number of data rather than the original chaotic attractors and thus, greatly aids to reveal their hidden self-similar nature.

This paper investigates the exponential stability of abstract mean field systems in their synchronized state. We analyze stability by studying the linearized system and demonstrate the existence of an exponentially stable invariant manifold. Our focus is on the equilibrium stability under synchronization. We provide a comprehensive analysis of both linear and nonlinear cases of the system. Additionally, we prove the existence of stable limit cycles and establish a relation between the dynamics in linear and nonlinear frameworks.

In this paper, we use the invariance method to obtain symmetries and exact solutions for a class of difference equations with variable coefficients. We look at the stability of the equilibrium points admitted by this class. Behavior and periodicity of their solutions are investigated. Consequently, results in existing literature [Elsayed, E.M., Alofi, B. S. and Khan,A. Q. (2022), Qualitative behavior of solutions of tenth-order recursive sequence equation, Mathematical Problems in Engineering, ID 5242325] are generalized.