Vol. 10, No. 3 (2021): Regular Issue

In this paper, we propose the SEIR epidemic model with delay and nonlinear incidence rate. The resulting model has two possible equilibria: if $R_{0} \leq 1,$ then the SEIR epidemic model has a disease-free equilibrium and if $R_{0} > 1,$ then the SEIR epidemic model admits a unique endemic equilibrium. By using suitable Lyapunov functionals and LaSalle's invariance principle, the global stability of a disease-free equilibrium is established. Our main contribution affirms the existence of non constant periodic solutions which bifurcate from the endemic equilibrium when the delay crosses some critical values. Finally, some numerical simulations are presented to illustrate our theoretical results.

We suggest a general variational like inequalities in an infinite dimensional space, where only approximation sequences are known instead of an exact value of the cost mapping and feasible set, and to apply a sequence of inexact solutions of auxiliary problems involving general penalty functions. Its convergence is attained without concordance of penalty, accuracy, and approximation parameters under certain coercivity type assumptions.

A switching strategy is also known as a Parrondo's paradox game in which the alternation of two dynamics may yield a desirable solution whereas it yielded an undesirable outcome individually. In this paper, the switching strategy is applied to the logistic map (LM) $x_{n+1}=rx_n (1-x_n)$ and its variants i.e. modified LM (Mod-LM) and extended LM (Ex-LM) in Ishikawa orbit. And the alternated LM and its complex forms in terms of the control parameter value, convergence point, and chaotic range of varying changeable parameters are analyzed The experimental results show that the switching theory succeeds to achieve the required outcome, i.e. ``undesirable + undesirable = desirable'' by getting the reduced extinction range and ``chaos + chaos = order'' by converting chaotic behavior to desirable oscillatory behavior. Also, a stability range analysis of Picard, Mann, and Ishikawa iterated logistic map and its complex form is conducted to study the behavior of each map in a different orbit.

This paper explores the qualitative analysis of a modified Leslie-Gower prey-predator model where the consumption rate of prey is by per capita predator according to Beddington-DeAngelis functional response. Moreover, time-lag $(\tau)$ is established to exploit a gestation period of predations. The permanence analysis of the proposed system is investigated. We study the local stability of the non-delayed model at all possible equilibrium points. It is demonstrated that the given model experiences Hopf bifurcation about the interior equilibrium point with respect to delay $\tau$. Thereafter the stability and direction of Hopf bifurcation are formulated through normal and center manifold theorems. The derived criteria are justified with the help of numerical simulations.

Fractional-order PD (PD$^{\lambda })$ control of friction-induced vibrations in a beam-mass model is analysed in this paper. Mathematical modelling of the system with a piezo-patch actuator bonded to the beam surface is presented. Linear stability analysis is performed to determine the stability boundary corresponding to the Hopf bifurcation points. The nature of bifurcation is found to be supercritical from the non-linear analysis by the method of averaging. The role of fractional-order on the effectiveness of the controller in quenching friction-induced vibrations is thoroughly investigated. The efficacy of the controller with varying size and locations of the piezo-patch is also studied.

The effects of a distributed `weak generic kernel' delay on cyclically coupled limit cycle and chaotic oscillators are considered. For coupled Van der Pol oscillators (and in fact, other oscillators as well) the delay can produce transitions from amplitude death(AD) or oscillation death (OD) to Hopf bifurcation-induced periodic behavior, with the delayed limit cycle shrinking or growing as the delay is varied towards or away from the bifurcation point respectively. The transition from AD to OD is mediated here via a pitchfork bifurcation, as seen earlier for other couplings as well. Also, the cyclically coupled undelayed van der Pol system here is already in a state of AD/OD, and introducing the delay allows both oscillations and AD/OD as the delay parameter is varied. This is in contrast to other limit cycle systems, where diffusive coupling alone does not result in the onset of AD/OD.\\ For systems where the individual oscillators are chaotic, such as a Sprott oscillator system or a coupled van der Pol-Rayleigh system with parametric forcing, the delay may produce AD/OD (as in the Sprott case), with the AD to OD transition now occurring via a transcritical bifurcation instead. However, this may not be possible, and the delay might just vary the attractor shape. In either of these situations however, increased delay strength tends to cause the system to have simpler behavior, streamlining the shape of the attractor, or shrinking it in cases with oscillations.

The main object of the present paper is to study the behavior of the approximated solutions of the nonlinear mixed Volterra-Fredholm integral equations by using Adomian Decomposition Method (ADM), Modified Adomian Decomposition Method (MADM), Variational Iteration Method (VIM) and Homotopy Analysis Method (HAM). Moreover, we discuss some new uniqueness results. Finally, some examples are included to demonstrate the validity and applicability of the proposed techniques.

This study deals with distinctive features of forming an effective potential for molecular ions and diatomic structures in three-dimensional radio-frequency traps. A simple model is proposed, which demonstrates the transition from vibration dynamics of micro-motion and micro-rotation to the time-averaged pseudopotential and rotation potential. It shows the existence of equilibrium states of a dimer molecule, which determine the stable orientation of an ion within the space of a three-dimensional Paul ion trap. Stable states and orbits for symmetrical and asymmetrical configurations were found.

We introduce a new relaxation function depending on an arbitrary parameter as a solution of a kinetic equation in the same way as the relaxation function introduced empirically by Debye, Cole-Cole, Davidson-Cole and Havriliak-Negami regarding, anomalous relaxation in dielectrics, which are recovered as particular cases. We propose a differential equation introducing a fractional operator written in terms of the Hilfer fractional derivative of order $\xi$, with $0 < \xi \leq 1$ and type $\eta$, with $0 \leq \eta \leq 1$. To discuss the solution of the fractional differential equation, the methodology of Laplace transform is required. As a by product we mention particular cases where the solution is completely monotone. %Some graphics show the behaviour of this completely monotone function. Finally, the empirical models are recovered as particular cases.

In this paper, we consider a prey-predator system allowing prey refuge and harvesting the prey species only. It is investigated under which condition the system has no limit cycle. An optimal harvesting policy is also formulated using Pontryagin's Maximum Principle. A comparison study have been done with the model in which the per individual prey refuge taken as constant. We investigate one and two parametric bifurcations thoroughly. Here we also discuss the bi-stability of the model in brief when the interior equilibrium is not unique. Some numerical simulations are given to verify our analytic works.

Landing gear shimmy is the name given to the wobbling motion due to torsional vibration and lateral flexing of a landing gear leg. Depending on the airplane requirements Main Landing Gears have different configurations. In this paper we analyze the sensitivity responses of the shimmy dynamics of a main landing gear for different geometric configurations. The dynamics are expressed in terms of three different degrees of freedom. We focus on the geometry of the main landing gear, specifically how the side-stay attachment point inclination angle affects shimmy.

An investigation of drug transport in the lumen as well as in the tissue, in the presence of absorbing interface is studied. The streaming blood is considered as a power-law fluid, whereas, the transport of luminal and tissue drug as a convection-diffusion process. Predicted results show the length of flow separation increases with increasing Re. Simulation predicts the diminishing tissue content with increasing wall absorption parameter. The luminal concentration reaches its quasi steady-state more rapidly than that in the tissue.

In this paper, we propose the global dynamics of an SIRSI epidemic model with discrete latent period and general nonlinear incidence function. By analyzing the corresponding characteristic equations, the local stability of the endemic equilibrium is established. By using suitable Lyapunov functionals and LaSalle's invariance principle, the global stability of the disease-free equilibrium and the endemic equilibrium are established for the SIRSI epidemic model with discrete latent period.

We study some results concerning dynamics and bifurcation of a special case of a second order rational difference equations with quadratic terms. We consider the second order, quadratic rational difference equation $$ x_{n+1} = \frac{\alpha+ \beta x_{n-1}}{A+B {x^2}_n+C x_{n-1}}, \ n=0,\ 1, \ 2, \ ... $$ with positive parameters $\alpha$, $\beta$, $A$, $B$, $C, $ and non-negative initial conditions. We investigate local stability, invariant intervals, boundedness of the solutions, periodic solutions of prime period two and global stability of the positive fixed points. And we study the types of bifurcation exist where the change of stability occurs. Then, we give numerical examples with figures to support our results.

The flow of non-Newtonian Giesekus fluid through a peristaltic annulus is presented. The interaction between the flow velocity and the heat transfer distributions is obtained due to the viscous dissipation in the energy equation. Due to the slow motion of the Giesekus fluid (Polymer solution), the small value of the Reynolds number is assumed. A modified longwave approximation, due to a series of the wavenumber, is applied to determine the zero as well as the first order distributions for the stress components, the velocity, and temperature. Also, the pressure rise, the radial position of the zero shear rates, and the rate of the heat flux are obtained numerically. The rheological properties of the non-Newtonian Giesekus fluid are discussed due to the mobility parameter and the time relaxation parameter. The graphical results illustrate that the mobility parameter and the time relaxation parameter enhance the flow. Meanwhile, the flow increases due to the peristaltic motion. According to the viscous dissipation effect, the temperature rises with the time relaxation parameter. Also, the contours of the streamlines are presented and the results illustrate that the trapping bolus disappears for the fluids with small-time relaxation and for the flow through a fixed channel without the peristalsis.