Инвариантные координатные подпространства нормальной формы системы обыкновенных дифференциальных уравнений

Abstract

For systems of ordinary differential equations (ODEs) with a nondegenerate linear part in the general and Hamiltonian cases, the problem of finding invariant coordinate subspaces in the coordinates of the normal form calculated in the vicinity of the equilibrium is stated. Conditions for the existence of such invariant subspaces in terms of the resonant relations between the eigenvalues of the linear part of the system are obtained. An algorithm for finding the resonant relations between the eigenvalues without their explicit calculation is described; this algorithm substantially uses computer algebra methods and the q-analog of the polynomial subresultants. The implementation of this algorithm in three popular computer algebra systems—Mathematica, Maple, and SymPy—is discussed. Interesting model examples are provided.