Numerical Approximation of the Spectra of Phan-Thien Tanner Liquids

The linear stability of the linear Phan-Thien Tanner (PTT) fluid model is investigated for plane Poiseuille flow. The PTT model involves parameters that can be used to fit shear and extensional data, which makes it suitable for describing both polymer solutions and melts. The base flow is determined using a Chebyshev-tau method. The linear stability equations are also discretized using Chebyshev approximations to furnish a generalized eigenvalue problem. The spectrum is shown to comprise a continuous part and a discrete part. The theoretical and numerical results are validated for the UCM and Oldroyd-B models, which are special cases of the PTT model, by comparing with results in the literature. It is demonstrated that the linear PTT fluid is stable to infinitesimal disturbances with respect to the range of shear-thinning, extensional and elasticity parameters considered. The computational efficiency and accuracy of the numerical method are also investigated.     

Local bifurcations of nonlinear viscoelastic panel in supersonic flow

The stability and bifurcation behaviors of a two-dimensional nonlinear viscoelastic panel in supersonic flow are investigated with analytical and numerical methods. One type of critical points for the bifurcation response equations is considered, which is characterized by a pair of purely imaginary eigenvalues and a pair of complex conjugate eigenvalues having negative real part. With the aid of computer language Maple and the normal form theory, Hopf bifurcation solution of the model is investigated. Finally, numerical simulations are shown, which agree with the theoretical analytical results.     

Analytical and numerical methods for the stability analysis of linear fractional delay differential equations

In this paper, several analytical and numerical approaches are presented for the stability analysis of linear fractional-order delay differential equations. The main focus of interest is asymptotic stability, but bounded-input bounded-output (BIBO) stability is also discussed. The applicability of the Laplace transform method for stability analysis is first investigated, jointly with the corresponding characteristic equation, which is broadly used in BIBO stability analysis. Moreover, it is shown that a different characteristic equation, involving the one-parameter Mittag-Leffler function, may be obtained using the well-known method of steps, which provides a necessary condition for asymptotic stability. Stability criteria based on the Argument Principle are also obtained. The stability regions obtained using the two methods are evaluated numerically and comparison results are presented. Several key problems are highlighted.     

Exponential Stability for Delayed Stochastic Bidirectional Associative Memory Neural Networks with Markovian Jumping and Impulses

In this paper, the problem of stability analysis for a class of delayed stochastic bidirectional associative memory neural network with Markovian jumping parameters and impulses are being investigated. The jumping parameters assumed here are continuous-time, discrete-state homogeneous Markov chain and the delays are time-variant. Some novel criteria for exponential stability in the mean square are obtained by using a Lyapunov function, Ito’s formula and linear matrix inequality optimization approach. The derived conditions are presented in terms of linear matrix inequalities. The estimate of the exponential convergence rate is also given, which depends on the system parameters and impulsive disturbed intension. In addition, a numerical example is given to show that the obtained result significantly improve the allowable upper bounds of delays over some existing results.     

Nonlinear evolution of the travelling waves at the surface of a thin viscoelastic falling film

The problem of hydrodynamic instability of a thin condensate viscoelastic liquid film flowing down on the outer surface of an axially moving vertical cylinder is investigated. In order to improve the accuracy of numerical results, the viscoelastic and heat transfer parameters have been included into the governing equations. Also, the analytical solutions are obtained by utilizing the long-wave perturbation method. The influence of some physical parameters is discussed in both linear and nonlinear steps of the problem. It has been revealed that the stability of the film flow is weakened when the radius of cylinder and the temperature difference are reduced. Moreover, it is found that the increment of down-moving motion of the cylinder can enhance the flow stability. Further, the thin film flow can be destabilized by the viscoelastic property. The results show that both supercritical stability and subcritical instability can take place within the film flow system given appropriate conditions. Moreover, the absence of Reynolds number leads to an obvious difference in the behavior of some physical parameters.     

Passive aerodynamics control of plasma instabilities

The possibility of using a smart-damping scheme to modify the dynamic responses of plasma oscillations governed by a two-fluid model is considered. The passive aerodynamics control strategy is used to address this issue. The control efficiency is found by analyzing the conditions satisfied by the control gain parameters for which, the amplitude of oscillations is reduced both in the harmonic and chaotic states. In the regular state, the analytical stability analysis uses for linear oscillations the Routh-Hurwitz criterion while the Whittaker method and Floquet theory are utilized for nonlinear harmonic oscillations. The stability boundaries in the control gain parameter space is derived. The agreement between the analytical and numerical results is good. In the chaotic states, numerical simulations are used to perform quenching of chaotic oscillations for an appropriate set of control parameters.     

Exponential Stability for Delayed Stochastic Bidirectional Associative Memory Neural Networks with Markovian Jumping and Impulses

In this paper, the problem of stability analysis for a class of delayed stochastic bidirectional associative memory neural network with Markovian jumping parameters and impulses are being investigated. The jumping parameters assumed here are continuous-time, discrete-state homogenous Markov chain and the delays are time-variant. Some novel criteria for exponential stability in the mean square are obtained by using a Lyapunov function, Ito’s formula and linear matrix inequality optimization approach. The derived conditions are presented in terms of linear matrix inequalities. The estimate of the exponential convergence rate is also given, which depends on the system parameters and impulsive disturbed intension. In addition, a numerical example is given to show that the obtained result significantly improve the allowable upper bounds of delays over some existing results.     

Positive numerical solution for a nonarbitrage liquidity model using nonstandard finite difference schemes

In this article, we construct a numerical method based on a nonstandard finite difference scheme to solve numerically a nonarbitrage liquidity model with observable parameters for derivatives. This nonlinear model considers that the parameters involved are observable from order book data. The proposed numerical method use a exact difference scheme in the linear convection‐reaction term, and the spatial derivative is approximated using a nonstandard finite difference scheme. It is shown that the proposed numerical scheme preserves the positivity as well as stability and consistence. To illustrate the accuracy of the method, the numerical results are compared with those produced by other methods. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 210‐221, 2014     

A Finite Difference Method for the Model of Wheezes

1.IntroductionInordertostudythepitchofwheezesinpatients,J.B.Grotbergandothershavegivenaclassofmathematicalmodelof.he....l1'2]:WherebandVaretheLaplaceoperatorandgradientoperator,respectively.TheCartesiancomponents(u,w)arethedimen-sionlessaxialfluidvelocityanddimensionlessverticalfluidvelocityrespectively.4(x,z)t)isthevelocitypotentialfunction,Pisthedi-mensionlessfluidDressuredeterminedfromtheunsteadyBernoul1iequation(1.3),Paisthesteadydrivingpressure,I.istheexternalpressure.M,Ai,B,gandTar…     

New delay-dependent global asymptotic stability criteria of delayed Hopfield neural networks

In this paper, the global asymptotic stability of Hopfield neural networks (HNNs) with delays is investigated by utilizing Lyapunov functional method and the linear matrix inequality (LMI) technique. Distinct difference from other analytical approaches lies in “linearization” of the neural network model, by which the considered neural network model is transformed into a linear time-variant system. Then, a process, which is called parameterized first-order model transformation, is used to transform the linear system. Novel criteria for global asymptotic stability of the unique equilibrium point of delayed HNNs are obtained. The results are related to the size of delays. The obtained results are less conservative and restrictive than those established in the earlier references. Two numerical examples are given to show the effectiveness of our proposed method.     

Robust stability criteria for uncertain neutral type stochastic system with Takagi–Sugeno fuzzy model and Markovian jumping parameters

In this paper, the robust stability for uncertain neutral stochastic system with Takagi–Sugeno (T–S) fuzzy model and Markovian jumping parameters (MJPs) are investigated. The jumping parameters considered here are generated from a continuous-time discrete-state homogeneous Markov process, which are governed by a Markov process with discrete and finite-state space. Some novel sufficient conditions are derived to guarantee the asymptotic stability of the equilibrium point in the mean square. By utilizing the Lyapunov–Krasovskii functional, stochastic analysis theory, some free weighting matrices and linear matrix inequality (LMI) technique, the upper bound of time-varying delay is obtained by using Matlab® control toolbox. Finally, some numerical examples are given to show the effectiveness of the obtained results.     

Synchronization of a four-dimensional energy resource system via linear control

Synchronization of a four-dimensional energy resource system is investigated. Four linear control schemes are proposed to synchronize energy resource chaotic system via the back-stepping method. We use simpler controllers to realize a global asymptotical synchronization. In the first three schemes, the sufficient conditions for achieving synchronization of two identical energy resource systems using linear feedback control are derived by using Lyapunov stability theorem. In the fourth scheme, the synchronization condition is obtained by numerical method, in which only one state variable controller is contained. Finally, four numerical simulation examples are performed to verify these results.     

Stability of implicit-explicit linear multistep methods for ordinary and delay differential equations

Stability properties of implicit-explicit (IMEX) linear multistep methods for ordinary and delay differential equations are analyzed on the basis of stability regions defined by using scalar test equations. The analysis is closely related to the stability analysis of the standard linear multistep methods for delay differential equations. A new second-order IMEX method which has approximately the same stability region as that of the IMEX Euler method, the simplest IMEX method of order 1, is proposed. Some numerical results are also presented which show superiority of the new method.      

Linear and nonlinear vibrations of fractional viscoelastic Timoshenko nanobeams considering surface energy effects

In this paper, the linear and nonlinear vibrations of fractional viscoelastic Timoshenko nanobeams are studied based on the Gurtin–Murdoch surface stress theory. Firstly, the constitutive equations of fractional viscoelasticity theory are considered, and based on the Gurtin–Murdoch model, stress components on the surface of the nanobeam are incorporated into the axial stress tensor. Afterward, using Hamilton's principle, equations governing the two-dimensional vibrations of fractional viscoelastic nanobeams are derived. Finally, two solution procedures are utilized to describe the time responses of nanobeams. In the first method, which is fully numerical, the generalized differential quadrature and finite difference methods are used to discretize the linear part of the governing equations in spatial and time domains. In the second method, which is semi-analytical, the Galerkin approach is first used to discretize nonlinear partial differential governing equations in the spatial domain, and the obtained set of fractional-order ordinary differential equations are then solved by the predictor–corrector method. The accuracy of the results for the linear and nonlinear vibrations of fractional viscoelastic nanobeams with different boundary conditions is shown. Also, by comparing obtained results for different values of some parameters such as viscoelasticity coefficient, order of fractional derivative and parameters of surface stress model, their effects on the frequency and damping of vibrations of the fractional viscoelastic nanobeams are investigated.     

机械化数学在转子动力学研究中的应用

基于机械化数学-吴文俊消去法,分别采用短轴承油膜力模型和Muszynska转子力学模型,对转子轴承系统中的动力学行为与稳定性进行了分析研究.具体分析时,采用吴文俊特征列概念和基于Maple软件的符号计算平台,对短轴承涡动参数进行了解析分析,以及试算构造出了Liapunov函数,并给出了转子系统运动稳定性条件.     

微分本构粘弹性轴向运动弦线横向振动分析的差分法

给出了微分本构粘弹性轴向运动弦线横向振动数值仿真的一种差分法．文中建立了具有微分本构的粘弹性运动弦线的横向振动模型;通过对系统的控制方程和本构方程在不同的分数节点离散,得到一种新的差分方法．利用这一方法,弦线振动方程的数值计算过程可以交替地显式进行,且有较小的截断误差和好的数值稳定性．与通用的方法比较,新的方法计算简单、方便．文中利用方程的不变量检验了数值结果的可靠性,并利用这一方法给出了一类弦线模型的参数振动分析．     

Nonlinear stability characterization of thin Newtonian film flows traveling down on a vertical moving plate

The paper presents both the linear and nonlinear stability theories for the characterization of thin Newtonian film flows traveling down along a vertical moving plate. The linear model is first developed to characterize the flow behavior. After showing the inadequacy of the linear model in representing certain flow characteristics, the nonlinear kinematics model is then developed to represent the system. The long-wave perturbation method is employed to derive the generalized kinematic equations with free film surface condition. The linear model is solved by using the normal mode method for three different, namely, the quiescent, up-moving and down-moving, moving conditions. Subsequently, the elaborated nonlinear film flow model is solved by the method of multiple scales. The modeling results clearly indicate that both subcritical instability and supercritical stability conditions are possible to occur in the film flow system. The effect of the down-moving motion of the vertical plate tends to enhance the stability of the film flow.     

Fixed-order robust compensators via min-max principle

An important (some say, the major) reason for using feedback control is the presence of uncertain parameters which are a natural part of any real dynamical model. In this paper, we consider uncertain constant parameters in a time-invariant linear plant and announce some new results concerning robust compensator synthesis. Using the min-max principle, we derive necessary conditions for fixed-order linear robust controllers assuring asymptotic stability or relative stability. These necessary conditions are an extension of the Lagrange multiplier method. This is achieved using a cost function based on the inverse of the so-called critical constraint. We present both matrix and polynomial versions; the latter allows controllers of fixed structure. We suggest a probability-one homotopy algorithm and solve some examples from the literature.The authors wish to thank Professor R. Bental, Faculty of Industrial Engineering at the Technion, for his suggestion to replace the cost function based on the inverse critical polynomial by a logarithmic function.     

LMI conditions for stability of impulsive stochastic Cohen–Grossberg neural networks with mixed delays

In this paper, the global asymptotic stability of impulsive stochastic Cohen–Grossberg neural networks with mixed delays is investigated by using Lyapunov–Krasovskii functional method and the linear matrix inequality (LMI) technique. The mixed time delays comprise both the multiple time-varying and continuously distributed delays. Some new sufficient conditions are obtained to guarantee the global asymptotic stability of the addressed model in the stochastic sense using the powerful MATLAB LMI toolbox. The results extend and improve the earlier publications. Two numerical examples are given to illustrate the effectiveness of our results.     

Lattice hydrodynamic model of pedestrian flow considering the asymmetric effect

The original lattice hydrodynamic model of traffic flow is extended to single-file pedestrian movement at middle and high density by considering asymmetric interaction (i.e., attractive force and repulsive force). A new optimal velocity function is introduced to depict the complex behaviors of pedestrian movement. The stability condition of this model is obtained by using the linear stability theory. It is shown that the modified optimal velocity function has a remarkable influence on the neutral stability curve and the pedestrian phase transitions. The modified Korteweg-de Vries (mKdV) equation near the critical point is derived by applying the reductive perturbation method, and its kink-antikink soliton solution can better describe the stop-and-go phenomenon of pedestrian flow. From the density profiles, it can be found that the asymmetric interaction is more efficient than the symmetric interaction in suppressing the pedestrian jam. The numerical results are consistent with the theoretical analysis.