Nonlinear Rayleigh-Taylor instability in magnetic fluids with uniform horizontal and vertical magnetic fields

A weakly nonlinear evolution of two dimensional wave packets on the surface of a magnetic fluid in the presence of an uniform magnetic field is presented, taking into account the surface tension. The method used is that of multiple scales to derive two partial differential equations. These differential equations can be combined to yield two alternate nonlinear Schrödinger equations. The first equation is valid near the cutoff wavenumber while the second equation is used to show that stability of uniform wave trains depends on the wavenumber, the density, the surface tension and the magnetic field. At the critical point, a generalized formulation of the evolution equation governing the amplitude is developed which leads to the nonlinear Klein-Gordon equation. From the latter equation, the various stability crteria are obtained.     

Formation of wavy nanostructures on the surface of flat substrates by ion bombardment

A popular mathematical model for the formation of an inhomogeneous topography on the surface of a plate (flat substrate) during ion bombardment was considered. The model is described by the Bradley-Harper equation, which is frequently referred to as the generalized Kuramoto-Sivashinsky equation. It was shown that inhomogeneous topography (nanostructures in the modern terminology) can arise when the stability of the plane incident wavefront changes. The problem was solved using the theory of dynamical systems with an infinite-dimensional phase space, in conjunction with the integral manifold method and Poincaré-Dulac normal forms. A normal form was constructed using a modified Krylov-Bogolyubov algorithm that applies to nonlinear evolutionary boundary value problems. As a result, asymptotic formulas for solutions of the given nonlinear boundary value problem were derived.     

非线性粘弹性柱的稳定性和混沌运动

研究了受轴向周期力作用的各向同性简支柱的动力学稳定性。假定粘弹性材料满足Lea-derman非线性本构关系。导出运动方程为非线性偏微分-积分方程,并利用Galerkin方法简化为非线性微分-积分方程。应用平均法进行了稳定性分析,并用数值结果进行验证。数值结果还表明系统可能存在混沌运动。     

Nonlinear stability and chaos in electrohydrodynamics

Using the method of multiple scales, the nonlinear instability problem of two superposed dielectric fluids is studied. The applied electric filed is taken into account under the influence of external modulations near a point of bifurcation. A time varying electric field is superimposed on the system. In addition, the viscosity and variable gravity force are considered. A generalized equation governing the evolution of the amplitude is derived in marginally unstable regions of parameter space. A bifurcation analysis of the amplitude equation is carried out when the dissipation due to viscosity and the control parameter are both assumed to be small. The solution of a nonlinear equation in which parametric and external excitations are obtained analytically and numerically. The method of generalized synchronization is applied to determine the equations that describe the modulation of the amplitude and phase. These equations are used to determine the steady state equations. Frequency response curves are presented graphically. The stability of the proposed solution is determined applying Liapunov's first method. Numerical solutions are presented graphically for the effects of the different equation parameters on the system stability, response and chaos.     

Nonlinear Marangoni instability in dielectric superposed fluids

Summary The nonlinear Marangoni instability of two dielectric superposed fluids is investigated. The system is stressed by a normal electric field such that it allows for the presence of surface charges at the interface. The method of multiple scale perturbations is used in order to obtain uniformly valid expansions. Two nonlinear Schrödinger equations describing the perturbed system are obtained. One of these equations is used to describe analytically and numerically the necessary conditions for stability and instability near the marginal state, while the other equation is used to obtain the nonlinear electrohydrodynamic cutoff wavenumber separating stable and unstable disturbances for the system.     

Optimization Approach to the Robustness of Linear Delay Systems

By using the Lyapunov equation approach and an improved Razumikhin-type theorem, this paper presents a new robust stability criterion for a linear system subject to delayed time-varying nonlinear perturbations. Then, by using a parameter optimization technique, an efficient algorithm is derived for determining a desirable matrix for the Lyapunov equation. As a consequence, less conservative robust stability bounds for the perturbed system are achieved. Numerical examples are included to demonstrate the effectiveness of the proposed approach.     

Nonlinear stability of an axial electric field: Effect of interfacial charge relaxation

We introduce a nonlinear perturbation technique to third order, to study the stability between two cylindrical inviscid fluids, subjected to an axial electric field. The study takes into account the relaxation of electrical charges at the interface between the two fluids. At first order, a linear dispersion relation is obtained. Analytical and numerical results for the overstability and incipient instability conditions are given. For perfect dielectric fluids, the electric field has a stabilizing influence, while for leaky dielectric fluids, the electric field can have either a stabilizing or a destabilizing influence depending on the conductivity and permittivity ratios of the two fluids. At higher order, a nonlinear dispersion relation (nonlinear Ginzburg–Landau equation) is derived, describing the evolution of wave packets of the problem. For leaky dielectric fluids near the marginal state, a nonlinear diffusion equation (nonlinear incipient instability) is obtained. For perfect dielectric fluids, two cubic nonlinear Schrödinger equations are obtained. One of these equations to determine a nonlinear cutoff electric field separating stable and unstable disturbance, whereas the other is used to analyze the stability of the system. It is found that the nonlinear stability criterion depending on the ratio of permittivity, Such effects can only be explained successfully in the nonlinear sense, as the linear analysis unsuccessful to inform about them.     

分数阶时变广义系统稳定性分析

基于Caputo分数阶导数,研究了分数阶时变广义线性系统和分数阶时变广义非线性系统的稳定性问题.首先利用相关不等式,给出了一个时变广义线性系统无脉冲且稳定的充分条件.然后,通过慢子系统来判断快子系统的变化,并利用Riccati方程,建立了分数阶时变广义非线性系统是渐近稳定的判定准则.最后,给出了算例和Simulink仿真结果,以说明结论的正确性.     

Solitary waves in dusty plasmas with variable dust charge and two temperature ions

Propagation of nonlinear waves in dusty plasmas with variable dust charge and two temperature ions is analyzed. The Kadomtsev–Petviashivili (KP) equation is derived by using the reductive perturbation theory. A Sagdeev potential for this system has been proposed. This potential is used to study the stability conditions and existence of solitonic solutions. Also, it is shown that a rarefactive soliton can be propagates in most of the cases. The soliton energy has been calculated and a linear dispersion relation has been obtained using the standard normal-modes analysis. The effects of variable dust charge on the amplitude, width and energy of the soliton and its effects on the angular frequency of linear wave are discussed too. It is shown that the amplitude of solitary waves of KP equation diverges at critical values of plasma parameters. Solitonic solutions of modified KP equation with finite amplitude in this situation are derived.     

Nonlinear Kelvin-Helmholtz instability of two miscible ferrofluids in porous media

The nonlinear theory of the Kelvin-Helmholtz instability is employed to analyze the instability phenomenon of two ferrofluids through porous media. The effect of both magnetic field and mass and heat transfer is taken into account. The method of multiple scale expansion is employed in order to obtain a dispersion relation for the first-order problem and a Ginzburg–Landau equation, for the higher-order problem, describing the behavior of the system in a nonlinear approach. The stability criterion is expressed in terms of various competing parameters representing the mass and heat transfer, gravity, surface tension, fluid density, magnetic permeability, streaming, fluid thickness and Darcy coefficient. The stability of the system is discussed in both theoretically and computationally, and stability diagrams are drawn.     

A new space–time discretization for the Swift–Hohenberg equation that strictly respects the Lyapunov functional

The Swift–Hohenberg equation is a central nonlinear model in modern physics. Originally derived to describe the onset and evolution of roll patterns in Rayleigh–Bénard convection, it has also been applied to study a variety of complex fluids and biological materials, including neural tissues. The Swift–Hohenberg equation may be derived from a Lyapunov functional using a variational argument. Here, we introduce a new fully-discrete algorithm for the Swift–Hohenberg equation which inherits the nonlinear stability property of the continuum equation irrespectively of the time step. We present several numerical examples that support our theoretical results and illustrate the efficiency, accuracy and stability of our new algorithm. We also compare our method to other existing schemes, showing that is feasible alternative to the available methods.     

The determining equations for the nonclassical method of the nonlinear differential equation(s) with arbitrary order can be obtained through the compatibility

In this paper, firstly we show that the determining equations of the (1+1) dimension nonlinear differential equation with arbitrary order for the nonclassical method can be derived by the compatibility between the original equation and the invariant surface condition. Then we generalize this result to the system of the (m+1) dimension differential equations. The nonlinear Klein–Gordon equation, the (2+1)-dimensional Boussinesq equation and the generalized Nizhnik–Novikov–Veselov equation serve as examples illustrating this method.     

大系统渐近稳定的一般判别定理

利用分解集结和向量 V 函数判别大系统的稳定性是一个广泛采用的有效方法.但过去一般限于集结成常系数线性比较方程,判定的只是指数稳定.本文提出一个非线性比较方程的构造定理,推广了 Bailey 方法,并且进一步推广了作者的前期工作,判定了大系统的非指数稳定.     

Nonlinear Kelvin-Helmholtz instability of two miscible ferrofluids in porous media

The nonlinear theory of the Kelvin-Helmholtz instability is employed to analyze the instability phenomenon of two ferrofluids through porous media. The effect of both magnetic field and mass and heat transfer is taken into account. The method of multiple scale expansion is employed in order to obtain a dispersion relation for the first-order problem and a Ginzburg–Landau equation, for the higher-order problem, describing the behavior of the system in a nonlinear approach. The stability criterion is expressed in terms of various competing parameters representing the mass and heat transfer, gravity, surface tension, fluid density, magnetic permeability, streaming, fluid thickness and Darcy coefficient. The stability of the system is discussed in both theoretically and computationally, and stability diagrams are drawn. Received: July 25, 2002; revised: April 16, 2003     

A finite difference scheme for the nonlinear time-fractional partial integro-differential equation

In this paper, a finite difference scheme is proposed for solving the nonlinear time-fractional integro-differential equation. This model involves two nonlocal terms in time, ie, a Caputo time-fractional derivative and an integral term with memory. The existence of numerical solutions is shown by the Leray-Schauder theorem. And we obtain the discrete L₂ stability and convergence with second order in time and space by the discrete energy method. Then the uniqueness of numerical solutions is derived. Moreover, an iterative algorithm is designed for solving the derived nonlinear system. Numerical examples are presented to validate the theoretical findings and the efficiency of the proposed algorithm.     

Kink–antikink density wave of an extended car-following model in a cooperative driving system

We propose an extended optimal velocity model applicable to cooperative driving control system by considering the headway of arbitrary number of cars that precede and the relative velocity. The stability condition of the extended model is obtained by using the linear stability theory. The modified Korteweg-de Vries (mKdV) equation is derived to describe the traffic behavior near the critical point by applying the nonlinear analysis. Thus the traffic jams can be described by the kink–antikink density wave which is the solution of the mKdV equation. The simulation results confirm the analytical results and show that the traffic jams are suppressed more efficiently with considering not only the headway of more vehicles ahead but also the relative velocity.     

Analysis of a modified two-lane lattice model by considering the density difference effect

In this paper, a modified lattice hydrodynamic model of traffic flow is proposed by considering the density difference between leading and following lattice for two-lane system. The effect of density difference on the stability of traffic flow is examined through linear stability analysis and shown that the density difference term can significantly enlarge the stability region on the phase diagram. To describe the phase transition of traffic flow, the Burgers equation and mKdV equation near the critical point are derived through nonlinear analysis. To verify the theoretical findings, numerical simulation is conducted which confirms that traffic jam can be suppressed efficiently by considering the density difference effect in the modified lattice model for two-lane traffic.     

Nonlinear Mathieu equation and coupled resonance mechanism

The method of multiple-scales is used to determine a third-order solution for a cubic nonlinear Mathieu equation. The perturbation solutions are imposed on the so-called solvability conditions. Solvability conditions in the non-resonance case yield the standard Landau equation. Several types of a parametric Landau equation are derived in the neighborhood of five different resonance cases. These parametric Landau equations contain a parametric complex conjugate term or a parametric second-order complex conjugate term or a parametric complex conjugate term as well as a parametric second-order term. Necessary and sufficient conditions for stability are performed in each resonance case. Stability criteria correspond to each parametric Landau equation and are derived by linear perturbation. Stability criteria for the non-trivial steady-state response are discussed. The analysis leads to simultaneous resonance. Transition curves are performed in each case. Numerical calculations are made for some transition curves to illustrate the coupled resonance regions, where the induced stability tongues within the instability tongues are observed. The amplitude of the periodic coefficient of Mathieu equation plays a dual role in the stability criteria for nonlinear Mathieu equation.     

Stability of implicit Runge-Kutta methods for nonlinear stiff differential equations

Under the assumption that an implicit Runge-Kutta method satisfies a certain stability estimate for linear systems with constant coefficientsl ₂-stability for nonlinear systems is proved. This assumption is weaker than algebraic stability since it is satisfied for many methods which are not evenA-stable. Some local smoothness in the right hand side of the differential equation is needed, but it may have a Jacobian and higher derivatives with large norms. The result is applied to a system derived from a strongly nonlinear parabolic equation by the method of lines.     

A parameter study of epitaxial crystal growth

A third order autonomous ordinary differential equation is studied that is derived from a mathematical model of epitaxial crystal growth on misoriented crystal substrates. The solutions of the ODE correspond to the traveling wave solutions of a nonlinear partial differential equation which is related to the Kuramoto–Sivashinsky equation. The fixed points, the periodic solutions, and the heteroclinic orbits of the ODE are analysed, and stability results are given. A variety of nonlinear phenomena are observed, including Gavrilov–Guckenheimer bifurcations, homoclinic bifurcations, and a cascade of period doublings.