THE CRITERION FOR THE ASYMPTOTIC STABILITY OF THE ZERO SOLUTION OF A DYNAMIC SYSTEM OF ODE ON A PLANE IN TERMS OF THE TOPOLOGICAL INDEX OF A SINGULAR POINT OF A VECTOR FIELD
The criterion for the asymptotic stability of the zero solution of a dynamic system of ODE in a plane with homogeneous polynomial right-hand sides of any odd power of a class of dynamical systems, for which the problem of asymptotic stability is algebraically solvable, is established in the work. The criterion reduces the problem of asymptotic stability to the problem of computation of the topological index of a singular point of some polynomial vector field, constructed on the basis of the original ODE system, that allows to use for solving of the problem of asymptotic stability different technique for computing the index. In particular, the conditions of asymptotic stability, which were obtained, brought by effective results for computing the index of a singular point to the formulas, which allow to find out the asymptotic stability as a result of a finite number of arithmetic operations and the function sign on the coefficients of polynomials in right-hand sides of equations of a system.