Local stability of cubic differential systems

Abstract

Using Einstein’s notation [4]), the polynomial differential systems of finite dimension n and of degree at most k with coefficients in a field k of characteristic zero can be written as dx dt = a + ajα1x α1 + ajα1α2x 1x2 + ajα1···αrx α1 · · · xr (1) where j, α1, αr ∈ {1, . . . , n}, 1 ≤ r ≤ k, and for j = 1, . . . , n and for 2 ≤ r ≤ k, the tensor ajα1···αr (1 time contravariant and r times covariant) is symmetric with respect to the lower subscripts and we consider the action of the general linear group GL(n, k) on the phase space k. In this work, starting from a minimal system of generators of the ideal of algebraic invariants of polynomial differential systems (1) we will develop an algorithmic method to study the local stability of a given cubic differential system with the help of Grobner bases and using the GL(n, k)-invariants of this cubic system [2]. Our algorithmic method is motivated by the important role of the invariant theory in the survey of polynomial differential systems and the numerous works on Groebner basis one of the main practical tools for solving algebraic systems and computing algebraic varieties. Keywords Polynomial differential systems,linear transformations, cubic differential systems,singular points, invariant, covariant, Grobner bases.