The inverse problem for singular perturbed system with many-sheeted slow surfaces

L. I. Kononenko

Vladikavkazskij matematičeskij žurnal, Tome 25 (2023) no. 3, pp. 81-88

Résumé

We consider a singularly perturbed system of ordinary differential equations with small parameter, which describes a problem of chemical kinetics. We examine the system by using the method of integral manifolds that serves as a convenient tool for studying multidimensional singularly perturbed systems of differential equations and makes it possible to lower the dimension of the system. The integral manifold consists of sheets; for the small parameter $\varepsilon=0$, it is a slow surface. We formulate the direct and inverse problems for the system. The direct problem is as follows: given the right-hand sides of the system, find a solution to the system or prove its existence. The inverse problem is to find the unknown right-hand sides of the system of differential equations from some data on a solution of the direct problem. First, we consider the degenerate case, in which the small parameter $\varepsilon$ equals zero, and some restrictions are imposed on the dimension of the slow and fast variables, on the class of the right-hand sides that are assumed polynomial (with degree $1$), and on the number of sheets of the slow surface. Then we pass to the nondegenerate case $\varepsilon\neq 0$. In the case of a single sheet of the slow surface, the existence and uniqueness theorem was previously proven for a solution of the inverse problem. In this paper, we consider a system whose slow surface consists of several sheets. We prove an existence and uniqueness theorem for a solution to such a system. The proof is based on the result previously obtained for a system whose slow surface consists of a single sheet.