On stability of critical points of quadratic differential equations in nonassociative algebras

Abstract

In this note we treat the stability of nonzero critical points of the dieren tial equation _ x = x 2 in a commutative real nonasso- ciative algebra. As our rst result we prove that if a critical point lies in some Peirce subspace with respect to a nonzero idempotent, it cannot be stable. This improves a previously known result due to Kinyon and Sagle. As a second result we show that there exists 2-dimensional algebra, with a nonzero critical point and a nontrivial idempotent, such that the critical point is stable, so that the additional assumption in our rst result cannot be completely lifted.