Glasnik Matematicki, Vol. 38, No.1 (2003), 19-27.

ON STABILITY OF CRITICAL POINTS OF QUADRATIC DIFFERENTIAL EQUATIONS IN NONASSOCIATIVE ALGEBRAS

Borut Zalar and Matej Mencinger

Abstract.   In this note we treat the stability of nonzero critical points of the differential equation x' = x² in a commutative real nonassociative algebra. As our first 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 first result cannot be completely lifted.

2000 Mathematics Subject Classification.   34A34, 17A99.

Key words and phrases.   Quadratic differential equation, nonassociative algebra, critical points, ray solutions, projections, nilpotents, Peirce subspaces.