Global stability of a virus dynamics model with Beddington–DeAngelis incidence rate and CTL immune response

In this paper, the global stability of virus dynamics model with Beddington–DeAngelis infection rate and CTL immune response is studied by constructing Lyapunov functions. We derive the basic reproduction number R ₀ and the immune response reproduction number R ⁰ for the virus infection model, and establish that the global dynamics are completely determined by the values of R ₀. We obtain the global stabilities of the disease-free equilibrium E ₀, immune-free equilibrium E ₁ and endemic equilibrium E ^∗ when R ₀≤1, R ₀>1, R ⁰>1, respectively.     

Combined effect of viscous dissipation and thermal radiation on fluid flows over a non-linearly stretched permeable wall

An analysis is presented for the steady non-linear viscous flow of an incompressible viscous fluid over a horizontal surface of variable temperature with a power-law velocity under the influences of suction/blowing, viscous dissipation and thermal radiation. Numerical results are illustrated by means of tables and graphs. The governing partial differential equations are converted into nonlinear ordinary differential equations by a similarity transformation. The effects of the stretching parameter n, suction/blowing parameter b, Prandtl number σ, Eckert number E_c(E_c ^* )E_{c}(E_{c}^{ *} ) and radiation parameter N _R are discussed. Two cases are studied, namely, (i) Prescribed surface temperature (PST case) and, (ii) Prescribed heat flux at the sheet (PHF case).     

A Summary of New Predictive High Frequency Thermo-Vibrational Models in Porous Media

In this paper, we consider the effect of mechanical vibration on the onset of convection in porous media. The porous medium is saturated either by a pure fluid or by a binary mixture. The importance of a transport model on stability diagrams is presented and discussed. The stability threshold for the Darcy–Brinkman case in the Ra _Tc -R and k _c -R diagrams is presented (where Ra _Tc , k _c and R are the critical Rayleigh number, the critical wave number and the vibration parameters, respectively). It is shown that there is a significant deviation from the Darcy model. In the thermo-solutal case with the Soret effect, the influence of vibration on the reduction of multi-cellular convection is emphasized. A new analytical relation for obtaining the threshold of mono-cellular convection is derived. This relation shows how the separation factor Ψ is related to the controlling parameters of the problem, Ψ = f (R, ε*, Le), when the wave number k → 0. The importance of vibrational parameter definition is highlighted and it is shown how, by using a proper definition for vibrational parameter, we may obtain compact relationship. It is also shown how this result may be used to increase component separation.     

Fractional-order PD control at Hopf bifurcations in a fractional-order congestion control system

In this paper, we address the problem of the bifurcation control of a delayed fractional-order dual model of congestion control algorithms. A fractional-order proportional–derivative (PD) feedback controller is designed to control the bifurcation generated by the delayed fractional-order congestion control model. By choosing the communication delay as the bifurcation parameter, the issues of the stability and bifurcations for the controlled fractional-order model are studied. Applying the stability theorem of fractional-order systems, we obtain some conditions for the stability of the equilibrium and the Hopf bifurcation. Additionally, the critical value of time delay is figured out, where a Hopf bifurcation occurs and a family of oscillations bifurcate from the equilibrium. It is also shown that the onset of the bifurcation can be postponed or advanced by selecting proper control parameters in the fractional-order PD controller. Finally, numerical simulations are given to validate the main results and the effectiveness of the control strategy.

     

Finite time stability analysis of PD fractional control of robotic time-delay systems

A finite time stability test procedure is proposed for robotic system where it appears a time delay in PD^α fractional control system. Paper extends some basic results from the area of finite time and practical stability to linear, continuous, fractional order time invariant time-delay systems given in state-space form. Sufficient conditions for this kind of stability, for particular class of fractional time-delay systems are derived.     

Scaling law in saddle-node bifurcations for one-dimensional maps: a complex variable approach

The study of transient dynamical phenomena near bifurcation thresholds has attracted the interest of many researchers due to the relevance of bifurcations in different physical or biological systems. In the context of saddle-node bifurcations, where two or more fixed points collide annihilating each other, it is known that the dynamics can suffer the so-called delayed transition. This phenomenon emerges when the system spends a lot of time before reaching the remaining stable equilibrium, found after the bifurcation, because of the presence of a saddle-remnant in phase space. Some works have analytically tackled this phenomenon, especially in time-continuous dynamical systems, showing that the time delay, τ, scales according to an inverse square-root power law, τ∼(μ−μ _c )^−1/2, as the bifurcation parameter μ, is driven further away from its critical value, μ _c . In this work, we first characterize analytically this scaling law using complex variable techniques for a family of one-dimensional maps, called the normal form for the saddle-node bifurcation. We then apply our general analytic results to a single-species ecological model with harvesting given by a unimodal map, characterizing the delayed transition and the scaling law arising due to the constant of harvesting. For both analyzed systems, we show that the numerical results are in perfect agreement with the analytical solutions we are providing. The procedure presented in this work can be used to characterize the scaling laws of one-dimensional discrete dynamical systems with saddle-node bifurcations.     

Zum zeitlichen Verlauf des Kriechens von Zementstein und Beton

Zusammenfassung Es wird eine Kriechfunktion abgeleitet, die die bekannten, empirisch bestimmten Kriechfunktionen als Spezialfälle enthält. Für die relative Endkriechverformung ergibt sich sowohl aus der exakten Lösung als auch aus den Näherungen der Wert 2/E. Für die Halbwertszeit, nach der die Hälfte des Endkriechmaßes erreicht wird, bekommt mann = 2l RZ , E. R ist eine Größe, die der inneren Reibung proportional ist. Es wird eine Beziehung angegeben, dieR mit der AktivierungsenergieQ verknüpft. Mit den aus Kriechkurven zu entnehmenden Werten fürR ist damit auchQ bekannt.

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Summary A formula for creep deformation as a function of time has been derived. It has been shown, that the empirical functions previously used by several authors are special cases of this more general function. The final creep deformation is found to be 2/E. Half of the final creep deformation is reached after a time ofn=21RZ/qE. R is a value, which is proportional to internal friction. There exists a relation betweenR and the activation energy. The activation energy can be calculated ifR is evaluated from creep data.

     

Stability analysis of linear fractional differential system with multiple time delays

In this paper, we study the stability of n-dimensional linear fractional differential equation with time delays, where the delay matrix is defined in (R ⁺)^n×n. By using the Laplace transform, we introduce a characteristic equation for the above system with multiple time delays. We discover that if all roots of the characteristic equation have negative parts, then the equilibrium of the above linear system with fractional order is Lyapunov globally asymptotical stable if the equilibrium exist that is almost the same as that of classical differential equations. As its an application, we apply our theorem to the delayed system in one spatial dimension studied by Chen and Moore [Nonlinear Dynamics 29, 2002, 191] and determine the asymptotically stable region of the system. We also deal with synchronization between the coupled Duffing oscillators with time delays by the linear feedback control method and the aid of our theorem, where the domain of the control-synchronization parameters is determined.     

Radiation effect on mixed convection over an isothermal cone in porous media

The radiation effect on the mixed convection flow of an optically dense viscous fluid adjacent to an isothermal cone embedded in a saturated porous medium with Rosseland diffusion approximation is numerically investigated. The entire regime of the mixed convection is included, as the mixed convection parameter of χ varies from 0 (pure free convection) to 1 (pure forced convection). The transformed nonlinear system of equations is solved by using an implicit finite difference method. Numerical results are given for the dimensionless temperature profiles and the local Nusselt number for various values of the mixed convection parameter χ, the cone angle parameter m, the radiation-conduction parameter R _d and the surface temperature parameter H. The local Nusselt number decreases initially, reaches a minimum in the intermediate value of χ and then increases gradually. It is apparent that increasing the cone angle parameter m enhances the local Nusselt number. The local Nusselt number is significantly increased for the large values of the radiation-conduction parameter R _d and the surface temperature parameter H, i.e., radiation effect becomes pronounced. Received on 25 October 1999     

Boundary effects on the Bénard-Marangoni instability under an electric field

This article theoretically studies the Bénard-Marangoni instability problem for a liquid layer with a free upper surface, which is heated from below by a heating coil through a solid plate in ana.c. electric field. The boundary effects of the solid plate, which include its thermal conductivity, electric conductivity and thickness, have great influences on the onset of convective instability in the liquid layer. The stability analysis in this study is based on the linear stability theory. The eigenvalue equations obtained from the analysis are solved by using the fourth order Runge-Kutta-Gill's method with the shooting technique. The results indicate that the solid plate with a higher thermal or electric conductivity and a bigger thickness tends to stabilize the system. It is also found that the critical Rayleigh numberR _c, the critical Marangoni numberM _c, and the criticala.c. Rayleigh numberE _ac become smaller as the intensity of thea.c. electric field increases.     

MHD mixed convective heat transfer flow about an inclined plate

Mixed convection heat transfer about a semi-infinite inclined plate in the presence of magneto and thermal radiation effects is studied. The fluid is assumed to be incompressible and dense. The nonlinear coupled parabolic partial differential equations governing the flow are transformed into the non-similar boundary layer equations, which are then solved numerically using the Keller box method. The effects of the mixed convection parameter R _i, the angle of inclination α, the magnetic parameter M and the radiation–conduction parameter R _d on the velocity and temperature profiles as well as on the local skin friction and local heat transfer parameters. For some specific values of the governing parameters, the results are compared with those available in the literature and a fairly good agreement is obtained.     

Auto-correlation measurements and integral time scales in three-dimensional turbulent boundary layers

Auto-correlation, time and length scales of the three components of turbulence and power spectra in a three-dimensional turbulent boundary layer developing on a yawed flat plate have been obtained. The measurements indicate that close to the wall, in the region of turbulence production, there is a marked disparity among the time scales but as the outer edge of the boundary layer is approached, the scales become comparable to one another. Also, the behaviour of the length scales and the power spectra across the boundary layer is presented.Nomenclature Boundary layer thickness where Q/Q _e=0.995 - E _u(f) one dimensional frequency spectra - f frequency in Hz - k ₁ wave number defined as k ₁=2f/Q - L length scale defined as: time scale times local mean velocity - Q local mean velocity - Q _e free stream velocity - R _u, R _v, R _w Auto-correlation coefficients of u, v and w respectively as defined in equation (1) - T _u, T _v, T _w the time scales of u, v and w fluctuations as defined in equation (2) - delay time - u fluctuating velocity component in x-direction - v fluctuation velocity component in y-direction - w fluctuation velocity component in z-direction - x coordinate axis in the streamwise direction - y coordinate axis normal to the surface - z coordinate axis normal to the x-direction and parallel to the wall     

Investigation of failure during elongational flow of polymer melts

A basic study of the mechanisms of necking and ductile failure of polymer melts in uniaxial elongational flow has been carried out. A linear stability analysis was carried out using a White—Metzner convected Maxwell model with a deformation-rate-dependent relaxation time, which varies according to τ = τ_o/(1 + aτ_o[2trd²]^{$\text{1}\text{2}$}). It was shown that filament stability and elongation to break depend upon τ_oE, where E is the elongation rate, and a. At fixed τ_oE, filament stability decreases with increasing a. At small a, stability increases with increasing τ_oE while for a > $\text{1}\text{√3}$, stability decreases with increasing τ_oE. For a material with small a, ductile failure can occur for small τ_oE, but cohesive fracture should be the cause of failure at larger τ_oE. For a material with large a, however, ductile failure always dominates the failure mode. These results are used to interpret failure in elongational flow of low density and high density polyethylene and polypropylene melts and describe how the latter two melts exhibit ductile failure.     

On the Convection in a Porous Medium with Inclined Temperature Gradient and Vertical Throughflow. Part I. Normal Modes

In this second part of our analysis of the destabilization of transverse modes in an extended horizontal layer of a saturated porous medium with inclined temperature gradient and vertical throughflow, we apply the mathematical formalism of absolute and convective instabilities to studying the nature of the transition to instability of such modes by assuming on physical grounds that the transition is triggered by growing localized wavepackets. It is revealed that in most of the parameter cases treated in the first part of the analysis (Brevdo and Ruderman 2009), at the transition point the evolving instability is convective. Only in the cases of zero horizontal thermal gradient, and in the cases of zero vertical throughflow and the horizontal Rayleigh number R _h < 49, the instability is absolute implying that, as the vertical Rayleigh number, R _v, increases passing through its critical value, R _vc, the destabilization tends to affect the base state throughout and eventually destroys it at every point in space. For the parameter values considered, for which the destabilization has the nature of convective instability, we found that, as R _v, increases beyond the critical value, while the horizontal Rayleigh number, R _h, and the Péclet number, Q _v, are kept fixed, the flow experiences a transition from convective to absolute instability. The values of the vertical Rayleigh number, R _v, at the transition from convective to absolute instability are computed. For convectively unstable, but absolutely stable cases, the spatially amplifying responses to localized oscillatory perturbations, i.e., signaling, are treated and it is found that the amplification is always in the direction of the applied horizontal thermal gradient.     

Threshold Solutions and Sharp Transitions for Nonautonomous Parabolic Equations on $${\mathbb{R}^N}$$

This paper is devoted to a class of nonautonomous parabolic equations of the form u _t = Δu + f(t, u) on \mathbbR^N{\mathbb{R}^N} . We consider a monotone one-parameter family of initial data with compact support, such that for small values of the parameter the corresponding solutions decay to zero, whereas for large values they exhibit a different behavior (either blowup in finite time or locally uniform convergence to a positive constant steady state). We are interested in the set of intermediate values of the parameter for which neither of these behaviors occurs. We refer to such values as threshold values and to the corresponding solutions as threshold solutions. We prove that the transition from decay to the other behavior is sharp: there is just one threshold value. We also describe the behavior of the threshold solution: it is global, bounded, and asymptotically symmetric in the sense that all its limit profiles, as t → ∞, are radially symmetric about the same center. Our proofs rely on parabolic Liouville theorems, asymptotic symmetry results for nonlinear parabolic equations, and theorems on exponential separation and principal Floquet bundles for linear parabolic equations.     

Transition from Convective to Absolute Instability in a Porous Layer with Either Horizontal or Vertical Solutal and Inclined Thermal Gradients,and Horizontal Throughflow

Recently, in Diaz and Brevdo (J Fluid Mech 681: 567–596, 2011), further in the text referred to as D&B, we found an absolute/convective instability dichotomy at the onset of convection in a flow in a saturated porous layer with either horizontal or vertical solutal and inclined thermal gradients, and horizontal throughflow. The control parameter in D&B triggering the destabilization is the vertical thermal Rayleigh number, R _v. In this article, we treat the parameter cases considered in D&B in which the onset of convection has the character of convective instability and occurs through longitudinal modes. By increasing the vertical thermal Rayleigh number starting from its critical value, R _vc, we determine the value R _vt of R _v at which the transition from convective to absolute instability takes place and compute the physical characteristics of the emerging absolutely unstable wave packet. In some cases, the value of the transitional vertical thermal Rayleigh number, R _vt, is only slightly greater than the critical value, R _vc, meaning that at the onset of convection the base convectively unstable state can be viewed as marginally absolutely unstable. However, in several cases considered, the value of R _vt is significantly greater than the critical value, R _vc, implying that the base state is not marginally but essentially absolutely stable at the point of destabilization.     

Temperature compensation of hot-wire anemometer with photoconductive cell

A new temperature compensation technique for hot-wire anemometer is proposed in this article. In contrast to the available compensation techniques, a photoconductive cell is introduced here as a variable resistor in the bridge. The major advantage of adopting an active component such as photoconductive cell is that temperature compensation can be achieved by using any kind of temperature sensors, once the output of temperature sensor is given as a voltage. Validation experiments using a photoconductive cell with a thermocouple-thermometer are conducted in the temperature range from 30 to 50 °C and the velocity ranges from 8 to 18 m/s.List of symbols h(U) convective heat transfer coefficient of the wire at U - E _b bridge top voltage - E _w voltage across the wire - I heating current through the hot-wire - R _A resistance connected in tandem with the hot-wire in the bridge - R _B variable resistance for overheat-ratio setting of the hot-wire - R _C compensation resistance connected in series with R _B in the bridge, 1/(R _p ^–1 + R _CdS ^–1 - R _CdS photoconductive cell resistance - R _P resistance connected parallel with R _CdS - R _wf wire resistance at T _f - R _wa wire resistance at T _a - R _ww wire resistance at working temperature of T _w - R _w0 wire resistance at 0 °C - T _f fluid temperature - T _a ambient temperature - T _w working temperature of the wire - U flow velocity - V _c compensating voltage applied to the input side of the R _CdS Greek symbols _w temperature resistance coefficient of the hot-wire - V voltage across R _A, voltage difference between E _b and E _w Greek symbols ^* reference     

Projective-dual synchronization in delay dynamical systems with time-varying coupling delay

The existence of projective-dual-anticipating, projective-dual, and projective-dual-lag synchronization in a coupled time-delayed systems with modulated delay time is investigated via nonlinear observer design approach. Transition from projective-dual-anticipating to projective-dual synchronization and from projective-dual to projective-dual-lag synchronization as a function of variable coupling delay τ _p (t) is discussed. Using Krasovskii–Lyapunov stability theory, a general condition for projective-dual synchronization is derived. Numerical simulations on the chaotic Ikeda and Mackey–Glass systems are given to demonstrate the effectiveness of the theoretical results.     

Asymptotic Axially Symmetric Deformations for Perfectly Elastic Neo-Hookean and Mooney Materials

For axially symmetric deformations of the perfectly elastic neo-Hookean and Mooney materials, formal series solutions are determined in terms of expansions in appropriate powers of 1/R, where R is the cylindrical polar coordinate for the material coordinates. Remarkably, for both the neo-Hookean and Mooney materials, the first three terms of such expansions can be completely determined analytically in terms of elementary integrals. From the incompressibility condition and the equilibrium equations, the six unknown deformation functions, appearing in the first three terms can be reduced to five formal integrations involving in total seven arbitrary constants A, B, C, D, E, H and k ², and a further five integration constants, making a total of 12 integration constants for the deformation field. The solutions obtained for the neo-Hookean material are applied to the problem of the axial compression of a cylindrical rubber tube which has bonded metal end-plates. The solution so determined is approximate in two senses; namely as an approximate solution of the governing equations and for which the stress free and displacement boundary conditions are satisfied in an average manner only. The resulting load-deflection relation is shown graphically. The solution so determined, although approximate, attempts to solve a problem not previously tackled in the literature.      

Stochastic resonance in a bias monostable system driven by a periodic rectangular signal and uncorrelated noises

The stochastic resonance in a bias monostable system driven by a periodic rectangular signal and uncorrelated noises is investigated by using the theory of signal-to-noise (SNR) in the adiabatic limit. The analytic expression of the SNR is obtained for arbitrary signal amplitude without being restricted to small amplitudes. The SNR is a nonmonotonic function of intensities of multiplicative and additive noises and the noise intensity ratio R=D/Q, so stochastic resonance exhibits in the bias monostable system. We investigate the effect of any system parameter (such as D,Q,R,r) on the SNR. It is shown that the SNR is a nonmonotonic function of the static asymmetry r, also; the SNR is decreased when |r| is increased. Moreover, the SNR is increased when the noise intensity ratio R=D/Q is increased.