The existence of a step-like contrast structure for a class of high-dimensional singularly perturbed system is shown by a smooth connection method based on the existence of a first integral for an associated system. In the framework of this paper, we not only give the conditions under which there exists an internal transition layer but also determine where an internal transition time is. Meanwhile, the uniformly valid asymptotic expansion of a solution with a step-like contrast structure is presented.

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On Step-Like Contrast Structure of Singularly Perturbed Systems

In this paper, the existence of step-like contrast structure for a class of singularly perturbed optimal control problem is shown by the contrast structure theory. By means of direct scheme of boundary function method, we construct the uniformly valid asymptotic solution for the singularly perturbed optimal control problem. Finally, an example is presented to show the result. 2000 Mathematics Subject Classification. 34B15; 34E15.

/pdf/step-like-contrast-structure-of-singularly-perturbed-optimal-1gn0zq5953.pdf

Step-like contrast structure of singularly perturbed optimal control problem

Singularly Perturbed Problems with Boundary and Interior Layers: Theory and Application

It is known that steady solutions of the hydromagnetic dynamo problem describe sufficiently well the Global Magnetic Patterns (GMP) in spiral galaxies. This paper is devoted to nonlinear quasi-steady state GMP in a thin disc and the stability of these patterns. It is shown that for a majority of spiral galaxies (with a flat rotation curve and with the disc thickness being a linear function of the radius) any stable or long-living axisymmetric GMP with reversals of magnetic field do not exist. For Milky Way it is shown that the analytical results are in a qualitative agreement with the observational information revealing several stable or quasi-steady reversals. The main mathematical instrument is the method of boundary functions of the theory of singular perturbations. The ratio of thickness of the galactic disc to its radius is used as an asymptotic (small) parameter.

Stability of contrasting solutions of nonlinear hydromagnetic dynamo equations and magnetic fields reversals in galaxies

In this paper we investigate the step-type contrast structure of a second-order semilinear differential equation with integral boundary conditions. The asymptotic solution is constructed by the boundary function method, and the uniform validity of the formal solution is proved by the theory of differential equalities. 2000 Mathematics Subject Classification. 34B15; 34E05; 34E20.

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A. B. Vasil'eva

Papers

Asymptotic of a solution of contrast-structure type