Glasnik Matematicki, Vol. 53, No. 2 (2018), 331-342.

NEAR-IDEMPOTENTS, NEAR-NILPOTENTS AND STABILITY OF CRITICAL POINTS FOR RICCATI EQUATIONS

Borut Zalar and Matej Mencinger

Faculty of Civil Engineering,Transportation Engineering and Architecture, University of Maribor, Smetanova 17, 2000 Maribor, Slovenia
e-mail: borut.zalar@um.si

Faculty of Civil Engineering,Transportation Engineering and Architecture, University of Maribor, Smetanova 17, 2000 Maribor,
Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana,
Center for Applied Mathematics and Theoretical Physics, University of Maribor, Mladinska 3, 2000 Maribor, Slovenia
e-mail: matej.mencinger@um.si

Abstract.   The paper introduces two algebraic concepts, near-idempotents and near-nilpotents associated to subspaces 𝒩 of critical points, which can be used to re-phrase a theorem due to Boujemaa, El Qotbi and Rouiouih on stability for the Ricatti equation, x'(t)=x(t)2, associated to algebra 𝒜 ≈ ℝd. Using this concepts their result corresponds to the case dim 𝒩=1. Our main results are a generalization of the above mentioned theorem to 𝒩 of arbitrary dimension and a counter-example which shows, even in the general setting, that the essential condition that critical points must be eigenvectors of a suitable multiplication operator cannot be omitted from the formulation due to Boujemaa et al.

2010 Mathematics Subject Classification.   34A34, 17A99

Key words and phrases.   Quadratic differential systems, non-associative algebra, singular points, stability, near-nilpotent, near-idempotent

Full text (PDF) (free access)

DOI: 10.3336/gm.53.2.06

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