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
Department of Basic Sciences, Faculty of Civil Engineering,
University of Maribor, Smetanova 17, 2000 Maribor, Slovenia
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.
Key words and phrases. Quadratic differential equation,
nonassociative algebra, critical points, ray
solutions, projections, nilpotents, Peirce subspaces.
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