Metastable Periodic Patterns in Singularly Perturbed Delayed Equations

We consider the scalar delayed differential equation e[(x)\dot](t)=-x(t) +f(x(t-1)){\epsilon\dot x(t)=-x(t)\,+f(x(t-1))}, where ${\epsilon\,{>}\,0}${\epsilon\,{>}\,0} and f verifies either df/dx > 0 or df/dx < 0 and some other conditions. We present theorems indicating that a generic initial condition with sign changes generates a solution with a transient time of order exp(c/e){{\rm exp}(c{/}\epsilon)}, for some c > 0. We call it a metastable solution. During this transient a finite time span of the solution looks like that of a periodic function. It is remarkable that if df/dx > 0 then f must be odd or present some other very special symmetry in order to support metastable solutions, while this condition is absent in the case df/dx < 0. Explicit e{\epsilon}-asymptotics for the motion of zeroes of a solution and for the transient time regime are presented.     

Singularly Perturbed System of Differential Equations with Regular Singularity

We construct asymptotic solutions of a singularly perturbed system of differential equations with regular singularity.     

Singularly Perturbed Ordinary Differential Equations with Nonautonomous Fast Dynamics

Embedding of nonautonomous dynamics in a skew-product flow is employed in the analysis of singularly perturbed equations, where the fast dynamics is time-varying. Uniform convergence of the slow dynamics and statistical convergence of the fast dynamics are established. The limits are characterized in terms of projections of invariant probability measures of the skew-product flow in which the fast dynamics is embedded. These invariant measures are generated by the limiting equations of the original time-dependent process.     

A Class of Singularly Perturbed Generalized Boundary Value Problems for Quasi-Linear Elliptic Equation of Higher Order

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Uniform Exponential Dichotomy and Continuity of Attractors for Singularly Perturbed Damped Wave Equations

For , we consider a family of damped wave equations , where − Λ denotes the Laplacian with zero Dirichlet boundary condition in L ²(Ω). For a dissipative nonlinearity f satisfying a suitable growth restrictions these equations define on the phase space semigroups which have global attractors A _η, . We show that the family , behaves upper and lower semicontinuously as the parameter η tends to 0⁺.     

Convergence of Global and Bounded Solutions of Some Nonautonomous Second Order Evolution Equations with Nonlinear Dissipation

In this paper we study the asymptotic behavior of solutions of the following nonautonomous wave equation with nonlinear dissipation.

$\left\{\begin{array}{ll} u_{tt}+\vert u_{t}\vert^{\alpha}u_{t}-\Delta u +f(u)=g(t,x),\quad{\rm in}\,\mathbb{R}_{+}\times\Omega,\\ \qquad\qquad u(t,x)=0,\quad\, {\rm on}\,\mathbb{R}_{+}\times\partial\Omega,\end{array}\right.$

where f is an analytic function, α is a small positive real and g(t, ·) tends to 0 sufficiently fast in L ²(Ω) as t tends to ∞.

We also obtain a general convergence result and the rate of decay of solutions for a class of second order ODE containing as a special case

$\left\{\begin{array}{ll} \ddot{U}(t)+\Vert\dot{U}(t)\Vert^{\alpha}\dot{U}(t)+\nabla F(U(t))=g(t),\quad t \in \mathbb{R}_+,\\ \qquad U(0)=U_{0}\,\in \mathbb{R}^{N},\quad\dot{U}(0)=U_{1}\in \mathbb{R}^{N}. \end{array}\right.$

     

Integration of Special Linear Equations of the Second Order

Under certain conditions on the coefficients, the Chazy equation with constant coefficients reduces to a second-order linear differential equation with six singular points. Investigating this equation with the use of the Schwarz derivative, we obtain linear equations in which some of these six singular points coincide. An integration procedure for these equations is considered. Their general solutions are obtained in explicit form.     

Second Order Abstract Neutral Functional Differential Equations

In this paper we are concerned with a class of second order abstract neutral functional differential equations with finite delay in a Banach space. We establish the existence of mild and classical solutions for the nonlinear equation, and we show that the map defined by the mild solutions of the linear equation is a strongly continuous semigroup of bounded linear operators on an appropriate space. We use this semigroup to establish a variation of constants formula to solve the inhomogeneous linear equation.     

One Singularly Perturbed Problem of Turbulent Gas Dynamics

The problem of the interaction of a Prandtl–Mayer wave with a shear layer is solved using the small parameter method for the case where the flow vorticity in the shear layer is small. A direct expansion is constructed and its inadequacy at large distances from the vortex layer is proved. The strained coordinate method is used to obtain a uniformly adequate expansion. It is shown that for certain velocity distributions in the shear layer, the characteristics in the reflected simple wave resulting from the interaction intersect each other and a shock arises in the flow. There coordinates of the shock origin and the function describing the shock shape are obtained.     

Periodic Solutions of Second Order Differential Equations with Discontinuous Nonlinearities

We consider the periodic solutions of

On Asymptotics of Solutions of Nonlinear Differential Equations of the Second Order

We study the problem of asymptotics of unbounded solutions of differential equations of the form y″ = α₀ p(t)ϕ(y), where α₀ ∈ {−1, 1}, p: [a, ω[→]0, +∞[, −∞ < a < ω ≤ +∞, is a continuous function, and ϕ: [y ₀, +∞[→]0, +∞[ is a twice continuously differentiable function close to a power function in a certain sense.__________Translated from Neliniini Kolyvannya, Vol. 8, No. 1, pp. 18–28, January–March, 2005.     

锅炉给水泵转子动力学参数的一、二阶摄动识别研究

锅炉给水泵的临界转速是非常重要的参数,与结构密切相关。为了准确地计算出给水泵的干临界转速(在空气中)和湿临界转速(在水中),必须知道转轴上的零部件对其弯曲刚度的加强作用。本文利用摄动传递矩阵法,结合实验室的试验数据,对某电站锅炉给水泵转子的动力学参数进行了一阶和二阶摄动识别。给出了给水泵转子上的热装叶轮、轴套对转子刚度产生加强作用的一、二阶摄动识别结果,得到了修正的给水泵转子系统动力学计算模型。由修正模型计算得到的转子系统固有频率与实测的转子系统固有频率吻合良好。指出一阶摄动识别的结果可以满足工程实际的精度要求。     

Flocking Dynamics of Singularly Perturbed Oscillator Chain and the Cucker-Smale System

In this note, we show how the one-dimensional Cucker-Smale system for “flocking” dynamics can be recovered as a singular perturbation limit of a chain of damped oscillators.     

A Direct Approach to Homoclinic Orbits in the Fast Dynamics of Singularly Perturbed Systems

Homoclinic orbits in the fast dynamics of singular perturbation problems are usually analyzed by a combination of Fenichel's invariant manifold theory with general transversality arguments (see Ref. 29 and the Exchange Lemma in Ref. 16). In this paper an alternative direct approach is developed which uses a two-time scaling and a contraction argument in exponentially weighted spaces. Homoclinic orbits with one last transition are treated and it is shown how -expansions can be extracted rigorously from this approach. The result is applied to a singularity perturbed Bogdanov point in the FitzHugh–Nagumo system.Supported by DFG Schwerpunktprogramm Ergodentheorie, Analysis und effiziente Simulation dynamischer Systeme..     

关于采用高阶Boussinesq方程求解波浪散射问题的注记

本文讨论了采用高阶Boussinesq方程模拟波浪散射时对基本速度变量位置的局部光滑处理方法。通过光滑局部基本速度变量的取值深度,减小其高阶导数项的量值、加快级数收敛速度进而改善模型方程求解深水波浪散射问题的能力。对于底部边界具有一阶导数不连续的情况,通过局部光滑．可以将基本速度变量取值深度尖角转化为圆角过渡,从而改善速度分布。对于其它任意变化的底部边界,为了减少高阶底坡导数项的影响,在曲率和高阶底坡导数项与斜率具有相同量级的情况下亦需要对基本速度变量的取值深度局部光滑。数值计算结果表明本文提出的光滑技术可以很好地改善Boussinesq方程模拟浅水波和深水波在斜坡地形上散射问题的能力。     

On the Start-Up Behavior and Steady-State Oscillation of Singularly Perturbed Harmonic Oscillators

In this paper we present an analytic methodology for the analysis of a class of electrical harmonic oscillators. We combine geometric methods with the theory of singularly perturbed systems, which we use as tool for reduced order modeling. So we are able to define an easy to use formula in order to reduce an oscillatory system to center manifold. Thus we get a model for the start-up behavior as well as for the steady-state oscillation of sinusoidal oscillators. Furthermore, we demonstrate our technique by means of the Clapp oscillator which is an important member of the Colpitts, Clapp and Pierce oscillator family.     

Higher Order Convergence Rates in Theory of Homogenization: Equations of Non-divergence Form

We establish higher order convergence rates in the theory of periodic homogenization of both linear and fully nonlinear uniformly elliptic equations of non-divergence form. The rates are achieved by involving higher order correctors which fix the errors occurring both in the interior and on the boundary layer of our physical domain. The proof is based on a viscosity method and a new regularity theory which captures the stability of the correctors with respect to the shape of our limit profile.     

The Asymptotic Behavior of Solution for the Singularly Perturbed Initial Boundary Value Problems of the Reaction Diffusion Equations in a Part of Domain

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