On singularly perturbed quasilinear systems

In this paper, the objective is to give sufficient conditions for the existence of solution of the nonlinear two-point boundary value problem(1.1). And we employ these results to consider the boundary layer phenomena of the quasilinear weakly coupled singularly perturbed system (DP)_q.     

A class of singularly perturbed reaction diffusion systems

A class of singularly perturbed reaction diffusion systems are considered. Under suitable conditions, using the comparison theorem the asymptotic behavior of solution for the initial boundary value problems is studied. Project supported by the National Natural Science Foundation of China     

Parametric stability of singularly perturbed nonlinear uncertain systems

The concept of parametric stability is used to analyze the stability properties of a singularly perturbed system with some parameters. The domain in the parameter space where the system is parametrically stable is found. The problem of control of a dc electric motor is solved as an example     

Double canard cycles in singularly perturbed planar systems

Nonlinear Dynamics - We study the bifurcations of slow-fast cycles with two canard points in singularly perturbed planar systems. After analyzing the local dynamics of two canard points lying on...     

Quenching in a class of singularly perturbed mechanical systems

In this paper we get some results about the dynamics of a class of mechanical systems under strong dissipation. We have obtained the stability of the semi-trivial solutions in this class, its asymptotic expansion, and the existence of hyperbolic periodic orbits and their stability was analysed. From these results we have showed the existence of quenching in a particular case.     

Boundary and angular layer behavior in singularly perturbed semilinear systems

Some authors employed the method and technique of differential inequalities to obtain fairly general results concerning the existence and asymptotic behavior, as ε→n~+, of the solutions of scalar boundary value problems In this paper, we extend these results to vector boundary value problems, under analogous stability conditions on the solution u=u(t) of the reduced equation O=h(t, u) Two types of asymptotic behavior are studied, depending on whether the reduced solution u(t) has or does not have a con tinuous first derivative in (a, b) leading to the phenomena of boundary and angular layers.     

Robust input-to-state stability of discrete-time singularly perturbed systems with nonlinear perturbation

Nonlinear Dynamics - This paper is concerned with the robust input-to-state stability (ISS) analysis and control of discrete-time singularly perturbed systems (DTSPSs) with nonlinear perturbations....     

An application of nonsymmetric Lax-Milgram lemma to nonassociated plasticity

Based on a general assumption for plastic potential and yield surface, some properties of the nonassociated plasticity are studied, and the existence and uniqueness of the distribution of incremental stress and displacement for work-hardening materials are proved by using nonsymmetric Lax-Milgram lemma, when the work-hardening parameter A>F/Q/–F/, Q/.     

Bifurcations and synchronization of singularly perturbed oscillators: an application case study

We present numerically obtained bifurcations of two dual singularly perturbed nonlinear oscillating circuits. Each of the circuits comprises two coupled sections: a nonlinear section with dissipation and a linear one with a small constant biasing source. The two dual oscillators show an interesting pattern of hierarchical oscillations that follow the Stern?CBrocot tree with intervals of synchronization yielding Arnold??s tongues and their corresponding devil??s-like staircases. Several numerical results are included.     

Singularly continuous spectrum of singularly perturbed operators

We propose a construction of a singularly perturbed self-adjoint operator with a given compact set in its singularly continuous spectrum. In particular, the set can be a fractal of prescribed type. We use the construction of a singularly perturbed operator à for a given self-adjoint operator A in a Hilbert space $\mathcal{H}$ that solves the eigenvalue problem Ãψ _i = λ _iψ_i for a countable set Λ = {λ _i} _i=1 ^∞ of real numbers λ _i ∈ ?¹, |λ _i| < ∞, and an orthonormal system of vectors {ψ _i}, i = 1, 2 …, under certain additional general conditions.     

Construction of asymptotic formulas for solutions of singularly perturbed degenerate systems of differential equations

We obtain an asymptotic solution of the Cauchy problem for a singularly perturbed degenerate system of differential equations in the case of a singular limit pencil of matrices. Translated from Neliniini Kolyvannya, Vol. 11, No. 3, pp. 408–420, July–September, 2008.     

Third-order nonlinear singularly perturbed boundary value problem

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Nonlinear singularly perturbed problems of ultra parabolic equations

A class of nonlinear singularly perturbed problem of ultra parabolic equations are considered. Using the comparison theorem, the existence, uniqueness and its asymptotic behavior of solution for the problem are studied.     

An exponentially fitted difference scheme for the hyperbolic-hyperbolic singularly perturbed initial-boundary value problem

In this paper we discuss,an initial-boundary value problem of hyperbolic type with firstderivative with respect to x.The asymptotic solution is constructed and its uniform validityis proved under weader compatibility conditions.Then we develop an exponentially fitteddifference scheme and establish discrete energy inequality.Finally,we prove that thesolution of difference problem uniformly converges to the solution of the original problem.     

On the problem of stabilization of motion for a parametric family of nonlinear singularly perturbed systems

For nonlinear uncertain singularly perturbed systems, we construct a control guaranteeing their absolute parametric stability and estimate the set of values of the parameters for which this property of the system is preserved.     

Numerical solutions for singularly perturbed semi-linear parabolic equation

In this paper,we discuss singularly perturbed semi-linear parabolic equations for one dimension and two dimension,we find numerical solutions by using both the line-method and the exact difference scheme on a special non-uniform discretization mesh.The uniform convergence in e of the first order accuracy is obtained.     

The asymptotic expansions of singularly perturbed boundary value problems

In this paper we study the singularly perturbed boundary value problem:εy″=f(t,y,ε),y(0)=ξ(ε),y(1)=η(ε).where εis a positive small parameter.In the conditions:f_(?)(0,y,0)≥m_0 ,f_(?)(l,y,0)≥m_0 and f_(?)f(t,y,ε)≥0 ,we prove the existences,and uniformly valid asymptotic expansions of solutions for the given boundary value problems,and hence we improve the existing results.     

Contrast structure for singular singularly perturbed boundary value problem

The step-type contrast structure for a singular singularly perturbed problem is shown. By use of the method of boundary function, the formal asymptotic expansion is constructed. At the same time, based on sewing orbit smooth, the existence of the step- type solution and the uniform validity of the asymptotic expansion are proved. Finally, an example is given to demonstrate the effectiveness of the present results.     

Asymptotic expansion of solutions of a singularly perturbed boundary-value problem

For linear singularly perturbed systems of ordinary differential equations, we construct an asymptotic expansion of a solution by using the method of boundary functions. Using pseudoinverse matrices and projections, we find all terms of the asymptotic expansion in the noncritical case.Translated from Neliniini Kolyvannya, Vol. 7, No. 2, pp. 155–168, April–June, 2004.