Vol. 9, No. 4 (2020): Regular Issue

In this paper, an SIR epidemic model with an infectious period and a non-linear Beddington-DeAngelis type incidence rate function is considered. The dynamics of this model depend on the reproduction number $R_0$. Accurately, if $R_0<1$, we show the global asymptotic stability of the disease-free equilibrium by analyzing the corresponding characteristic equation and using comparison arguments. In contrast, if $R_0>1$, we see that the disease-free equilibrium is unstable and the endemic equilibrium is permanent and locally asymptotically stable and we give sufficient conditions for the global asymptotic stability of the endemic equilibrium.

In this work, we use the coordinate wise variational method for optimal resource allocation problems with involve simplex type constraints. It consists in making coordinate wise steps together with special threshold control and tolerance whose values reduce sequentially, and we establish the convergence rate under mild conditions.

Nonlinear systems state estimation has been an active research topic for some decades, since most nonlinear dynamics control algorithms require complete knowledge of the dynamic states and measuring all the states is often unfeasible. This note reviews former results and introduces a new methodology for determining the gains of a Luenberger-like observer employed with Lipschitz nonlinear dynamics. Some examples are presented to illustrate the use of the proposed algorithm and to compare it to recent solutions. This new methodology results in lower gains and enables the designer to determine the eigenvalues of the linear observer. It is shown that if the dynamics are in canonical form, this methodology can cope with a Lipschitz constant of any value.

We investigate the dynamics of a delay differential coupled Duffing-Van der Pol oscillator equation. Using the Lindstedt's method, we derive the in-phase mode solutions and then obtain the slow flow equations governing the stability of the in-phase mode by employing the two variable perturbation method. We solve the slow flow equations using series expansion and obtain conditions for Hopf bifurcation and studied stability of the in-phase mode. Finally, we studied stability and bifurcations of the origin. Our interest in this system is due to the fact that it is related to the coupled laser oscillators.

The authors introduce the concept of $(s,t)$-Volterra stochastic quadratic operator in a bisexual population, where each individual belongs either to the male sex group or female sex group. Under some conditions affecting the coefficients of these operators, several Lyapunov functions have been constructed so that the upper bounds for the set of limiting points of the trajectories could be obtained. This study depicts that the set of $(s,t)$-Volterra stochastic quadratic operators is a convex compact set and the extreme points of this set are found. Furthermore, $(s,t)$-Volterra stochastic quadratic operators of the aforementioned population, which have periodic trajectories, are constructed.

This work explores the influence of domain-size on the evolution of pattern formation modelled by an \textit{activator-depleted} reaction-diffusion system on a flat-ring (annulus). A closed form expression is derived for the spectrum of the Laplace operator on the domain. Spectral method is used to depict the close form solution on the domain. The bifurcation analysis of \textit{activator-depleted} reaction-diffusion system is conducted on the admissible parameter space under the influence of domain-size. The admissible parameter space is partitioned under a set of proposed conditions relating the reaction-diffusion constants with the domain-size. Finally, the full system is numerically simulated on a two dimensional annular region using the standard Galerkin finite element method to verify the influence of the analytically derived domain-dependent conditions.

This paper is devoted to design discrete PID controller for hyperchaos multi-switching synchronization. The dominant pole placement problem with such discrete PID controllers in z-domain is studied since it is important to take advantage of discrete domain representation, especially, during the pole placement procedure. Moreover, it is shown that modified Nyquist plot method is still valid in discrete domain and it is possible to find relevant discrete PID controller parameters. Then, the feasibility as well as the performance of the proposed approach of multi-switching combination synchronization, based on PID controller, is checked through its practical application in information transmission field to ensure more security of the message signal by means of hyperchaotic masking. Finally, experimental simulations are carried out in order to assess the security analysis and demonstrate that the suggested cryptosystem is large enough to resist to the noise attack thanks to its excellent encryption robustness.

This article explores the influence of non-Darcy porous medium on nanoliquid flow past a stretching plane with thermal radiation and thermal slip on surface. The pedesis and thermophoresis effects have been included into the nanofluid model. The equations governing the momentum and energy are initially cast into dimensionless form using the suitable non-dimensional variables. The resultant system of equations is then solved employing variational Finite Element Method (FEM). The impact of pertinent parameters such as thermal radiation, temperature jump, non-Darcy on the flow, temperature and local skin friction coefficient, local Nusselt number are investigated and presented graphically. The Forchheimer number is found crucial to increase temperature and lower the local heat transfer rate in the boundary layer.

The aim of this paper is to study the existence and uniqueness for nonlinear fractional differential equations involving Caputo's fractional derivative using the Krasnoselskii and Banach fixed point theorems on one hand and to establish the Ulam stability on the other hand. Finally, An example is given to substantiate the usefulness of the obtained results.

This article describes a mathematical model for a predator-prey system with two types of functional responses and a transmissible disease in the predator species. The effect of stocking of healthy predator in healthy predator species class and harvesting of healthy predator are analyzed. The essential mathematical features of the proposed eco-epidemic system such as boundedness, positivity, local and global stability and Hopf bifurcation are discussed. Numerical simulations are also performed to validate the theoretical results obtained.

We present in this article the existence and uniqueness results for a fractional functional differential equation with boundary condition and finite delay involving $\psi-$ Hilfer --type fractional derivative. Next, we establish the equivalent mixed-type integral for boundary condition. Further, the Ulam-Hyers-Mittag-Leffler stability is discussed. The Picard operator method, Banach fixed point theorem, and generalized Gronwall's inequality plays an important role to prove our results. At the end, an illustrative example will be introduce to justify our results.