Vol. 13, No. 3 (2024): Regular Issue

The geometry of orbits of families of smooth vector fields was studied by many mathematicians due to its importance in applications, in the theory of optimal control of dynamic systems, in geometry, and in the theory of foliations. In this paper it is studied geometry of orbits of vector fields in four dimensional Euclidean space. It is shown that orbits generate singular foliation ever regular leaf of which is a surface of negative Gauss curvature and zero normal torsion. In addition, the invariant functions of the considered vector fields are used to find solutions of the two-dimensional heat equation that are invariant under the groups of transformations generated by these vector fields. In the present paper, smoothness is understood as smoothness of the class $C^{\infty } $.

This research paper examines the problem of stabilizing inhomogeneous semilinear control systems in Hilbert state space. The paper proposes a feedback control that can achieve both strong and weak stabilization under certain assumptions related to approximate observability. The provided applications of the proposed method include the nonlinear Schroedinger equation and heat equation.

In this article, the mechanical analysis of Vallis system has been studied. Firstly, the Vallis system has been transformed into Kolmogorov type system, which is decomposed into four types of torques: inertial torque, internal torque, dissipation and external torque. Five scenarios are examined using combinations of various torques in order to identify the key elements in chaos creation and their physical significance. In these five scenarios, the conversion between kinetic energy, potential energy, and Hamiltonian energy is examined. It is examined how the energy and the parameters are interacting. The study comes to the conclusion that any combination of three forms of torques cannot create chaos in a Vallis system, and that a combination of these four types of torques is required to do so.

In this paper, an analytical model for single-celled, steady, incompressible and axisymmetric atmospheric vortices is presented. The velocity components and pressure are derived by substituting the assumed special form of the radial dependent azimuthal velocity of Wood and White [1] model into the governing equations. The azimuthal velocity component, vertical velocity component and pressure depend on radial and vertical coordinates whereas the radial velocity component depend only on the radial coordinate. The separation of variables method is applied for the solution of governing equations. This analytical model is then used to study the velocities and pressure of tornado-like stationary vortex. In this new approach, viscosity affects velocities as well as the pressure gradient. The radial velocity component decreases in magnitude as the Reynolds number increases. It is observed that the maximum azimuthal velocity weakens with altitude. The vertical profile of the azimuthal velocity increases up to some height and, once it attains maximum velocity, it start weakens gradually from the height of the maximum velocity and reduces zero asymptotically at higher altitudes. The peak of the radial pressure gradient decays with increasing altitude and has insignificant variation at higher altitudes. It is also observed that the axial pressure gradient falls with increasing Reynolds numbers.

The current study investigates vortex-induced vibration analysis of bladeless wind turbines made of functionally graded materials (FGMs). The bladeless wind turbine is modeled as a clamped-free cantilever beam with a circular cross-section. The mechanical properties of the turbine are assumed to be graded along the length of the beam according to the power-law distribution. The distributed aerodynamic force on the turbine is modeled based on an oscillator semi-empirical model. In order to analyze free vibrations, the functionally graded (FG) turbine's natural frequencies are calculated by the Ritz method and compared to the results obtained by Abaqus FEA. Good agreement is observed between analytical results and numerical values. Using Hamilton's principle, the governing dynamics equation of the FG turbine is derived based on the Euler-Bernoulli beam theory. The partial differential equations of motion are transformed into the ordinary differential equations employing the Galerkin method. A set of coupled differential equations are then solved by the Runge-Kutta method. Eventually, the effects of some parameters of the system, such as the turbine's length, turbine's cross-section, power-law gradient index, and wind velocity, on the system's dynamic response are discussed. The results show that the gradient index and geometric ratios significantly affect the wind turbine's vibration.

In 2015, south and north American countries witnessed an epidemic of Zika fever caused by the Zika virus (ZIKV). The virus got transmitted to humans primarily by the bite of aedes mosquitoes. Brazil and Colombia were among the countries which suffered the most during this Zika pandemic that lasted for almost three years. This paper introduces a non-linear extended SIR model to model the Zika pandemic, where two separate populations, humans and mosquitoes, are considered. Official data provided by health agencies drag our attention toward a possible relationship between environmental pollution and Zika infections. To model the effect of pollution, we have incorporated a stressed compartment of the human population that consists of those exposed to environmental pollution. We have derived the expression and actual value of Colombia's basic reproduction number of the Zika outbreak. Also, we have derived the conditions under which the disease-free and endemic equilibrium points of the model become stable and unstable, respectively. We have also shown whether the model will show backward bifurcation or not. A detailed qualitative analysis of the model has been done. We have conducted comprehensive numerical simulations to support our theoretical findings. Lastly, we investigated a massive Zika virus outbreak in Colombia (2015-17) with the help of the proposed model. The impact of environmental pollution has also been studied in the present study. Our work is the first official data-driven establishment between ecological contamination and the spread of the Zika outbreak.

In this manuscript, we primarily concentrate on the analysis of oscillatory behavior for the fractional order partial differential equation with damping term under Robin and Dirichlet boundary conditions. We obtained some new oscillation results by using the integral averaging technique and the generalized Riccati transformation. In the end, we have given two primary examples to illustrate the effectiveness of the obtained theory.

Anthropogenic activity along with natural activity has increased the level of pollutants into the atmosphere. The amount of pollutants has replaced the clean air molecules with the toxic air pollutants. This impacts the average rain fall negatively. Moreover, natural rain fall is one of the remedies to reduce toxicity. Toxicity in turn creates pollution. This cycle of pollutants, rain and toxicity is studies through the formulation of system of non-linear differential equations. Threshold for pollutants is performed to monitor the effect on rain and toxicity. To maintain threshold under control, backward bifurcation is workout. The results derived in the proposed model are supported by numerical simulation.

In this article, the Laplace Adomian Decomposition Method (LADM) is developed to investigate an approximate solution of nonlinear RLC Circuit Equation. This technique involving a permutation between the Laplace transforms (LTM) method with the Adomian decomposition method (ADM). The nonlinear part is involved through an infinite series of Adomian polynomials, and the (LADM) gives an infinite series solution to the equation.

In this article, we study the reachability of linear and non-linear fractional dynamical systems with single delay in control in the sense of $\psi$-Hilfer pseudo-fractional derivative. The necessary and sufficient conditions for reachability of linear fractional dynamical systems are obtained using Grammian matrix which is expressed by the Mittag-Leffler functions (one or two parameters). Sufficient conditions for reachability of nonlinear fractional dynamical systems are obtained by using Schauder's fixed point theorem. Two numerical examples are offered to help better understand of theoretical results.

This research investigates an SVIR epidemic model with a time delay that represents the latency period and immigration into all classes. The presence of Immigration will eliminate the disease-free equilibrium, and then there is no extinction scenario of the epidemic, and we deduce that immigration will eliminate the notion of threshold dynamics, and hence there is no basic reproduction number. We obtained that the epidemic is always persistent and the unique endemic equilibrium is globally stable, which has been proved using the Lyapunov approach.

HIV is a viral pathogen that weakens a human's immune organ, making it vulnerable to infectious diseases. This study focuses on a nonlinear deterministic mathematical model for the impact of HIV pathogens on AIDS disease transmission with optimal control analysis. Equilibrium points and basic reproduction number are computed. The qualitative analysis of the model revealed the scenario for both HIV-free and endemic equilibrium points. The local stability of the equilibrium is established via the Routh-Hurwitz criteria, while the global stability of the equilibrium is justified by using a Lyapunov function. Also, the normalized sensitivity analysis is performed. We extended the proposed model into an optimal control problem by incorporating four control variables, namely, a safer sex programme, a preventive measure, a condom usage programme, and medical care. Furthermore the optimal control is found by minimizing the number of HIV/AIDS individuals. Finally, the numerical simulations show agreement with the analytical results.

In this paper complete controllability of nonlinear neural control network systems is discussed. Using suitable substitution assumed system is converted in the first-order nonlinear differential equation with a nonzero linear part. With the use of controllability Grammian matrix, Lipschitz type nonlinearity, and fixed point theorem, some sufficient conditions for the complete controllability are derived. In the end, one numerical example and one LR (inductance and resistance) circuit example are discussed to validate the theoretical results.

A criterion based on generalized synchronization is provided for the extension of unpredictable oscillations among coupled Hopfield neural networks (HNNs). It is shown that if the drive network possesses an unpredictable oscillation, then the same is true for the response network provided that they are synchronized in the generalized sense. Extension of unpredictability in coupled 4D HNNs is exemplified with simulations. The auxiliary system approach and conditional Lyapunov exponents are utilized to demonstrate the presence of synchronization.

In this study, the solutions are provided for steady, incompressible, and electrically conducting non-Newtonian fluid flow over a stretching surface embedded in a porous medium with variable viscosity in the presence of a uniform transverse magnetic field, a heat source/sink, and Joule heating. With the Scaling group transformations, we transformed the non-linear coupled partial differential equations into coupled ordinary differential equations. Symbolic algebra software Maple was used to illustrate the effects of embedded variables on the distribution of velocity and heat functions, and a comparison of skin friction and temperature gradient was conducted using a table.