Stability of critical points of quadratic homogeneous dynamical systems

Abstract

Abstract. In this work, we give sufficient conditions ensuring the instability of a critical point of a homogeneous quadratic system in R n using the multiplication of the corresponding non-associative algebra. This result generalizes a theorem of Zalar and Mencinger (see [5]). We also state a classification theorem giving the stability or the instability of any stationary point of a quadratic homogeneous system in R 2 . As expected, the second theorem in [5] is part of this classification.

Near-idempotents, near-nilpotents and stability of critical points for Riccati equations

“…In the sequel we combine some of our previous work and Theorem 2.2 of [4], in order to obtain some interesting observations for the smallest possible case, i.e. dim (A) = 2 and dim (N ) = 1.…”

“…In the third section we prove our main result which is a natural extension of Boujemaa-El Qotbi-Rouiouih theorem to general subspaces of singular points and provide a counter-example in R 3 , in which the subspace of singular points is 2-dimensional, whose purpose is to show that the additional assumption of [23, Theorem 2.1] and [4], i.e. that singular points must belong to some eigenspaces of suitable algebraic elements, cannot be omitted.…”

“…For 3-dimensional case only partial results are known. In [ 4]) in two directions. They first proved that the above mentioned result remains true even when p is not necessarily an idempotent but an element satisfying a weaker algebraic condition, and then proceed to completely solve the problem when a singular point (i.e.…”

“…In this paper we define a general algebraic framework in which the first main result of [4] can be interpreted as a special case where the subspace of singular points is 1-dimensional. The second section will provide a clarification of 2-dimensional case in this new terminology.…”

On unbounded polynomial dynamical systems

“…In their paper on nonassociatice algebras and homogeneous quadratic differential equations (see [5]), Kinyon and Sagle gave substantial results and, more recently, this approach has been used for studying stability questions (see [1]). This process can be generalized to homogeneous polynomial differential equations of degree m ≥ 2.…”