TOPOLOGICAL ROUGHNESS OF DEGENERATE DYNAMICAL SYSTEMS AND HIGH-ORDER ROUGHNESS ESTIMATES

Abstract

In the article, on the basis of the method of topological roughness, questions of the roughness of degenerate dynamical systems and estimates of the roughness of a higher order are considered. The method of topological roughness of dynamical systems, which is based on the concept of roughness according to Andronov-Pontryagin, involves the study of the roughness of nonlinear systems near singular trajectories in the phase space, especially near singular points. In the traditional consideration of roughness at singular points, the matrices of the linear part are non-degenerate and the problem of calculating the roughness index in the form of the condition number of the matrix of reduction to a diagonal (quasi-diagonal) form is not difficult. But in practice, there are cases of degenerate matrices of the linear part for which it is proposed to use the procedures of matrix pseudo-inversion. Also, for essentially nonlinear systems, high-order roughness indicators are introduced based on matrix pseudo-inversion procedures.