#### 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

*e-mail:* `borut.zalar@uni-mb.si`

*e-mail:* `matej.mencinger@uni-mb.si`

**Abstract.** In this note we treat the stability of
nonzero critical points of the differential equation
*x*' = *x*^{2} 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.

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