Original scientific paper

https://doi.org/10.3336/gm.57.2.06

On a generalization of some instability results for Riccati equations via nonassociative algebras

Hamza Boujemaa orcid.org/0000-0001-8117-7227 ; Département de Mathématiques, Faculté des Sciences de Rabat, Mohammed V University in Rabat, 1014RP Rabat, Rabat, Morocco
Brigita Ferčec orcid.org/0000-0003-0990-2775 ; Faculty of Energy Technology, University of Maribor, Hočevarjev trg 1, 8270 Krško, Slovenia

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Abstract

In [28], for any real non associative algebra of dimension \(m\geq2\),
having \(k\) linearly independent nilpotent elements \(n_{1}\), \(n_{2}\), …,
\(n_{k},\) \(1\leq k\leq m-1\), Mencinger and Zalar defined near idempotents and
near nilpotents associated to \(n_{1}\), \(n_{2}\), …, \(n_{k}\). Assuming
\(\mathcal{N}_{k}\mathcal{N}_{k}=\left\{ 0\right\}\), where \(\mathcal{N}
_{k}=\operatorname*{span}\left\{ n_{1},n_{2},\ldots,n_{k}\right\} \), they
showed that if there exists a near idempotent or a near nilpotent, called \(u\),
associated to \(n_{1},n_{2},\ldots,n_{k}\) verifying \(n_{i}u\in\mathbb{R}n_{i},\)
for \(1\leq i\leq k\), then any nilpotent element in \(\mathcal{N}_{k}\) is
unstable. They also raised the question of extending their results to cases
where \(\mathcal{N}_{k}\mathcal{N}_{k}\not =\left\{ 0\right\} \) with
\(\mathcal{N}_{k}\mathcal{N}_{k}\subset\mathcal{N}_{k}\mathcal{\ }\)and to cases
where \(\mathcal{N}_{k}\mathcal{N}_{k} \not\subset \mathcal{N}_{k}.\)
In this paper, positive answers are emphasized and in some cases under the
weaker conditions \(n_{i}u\in\mathcal{N}_{k}\). In addition, we characterize all
such algebras in dimension 3.

Keywords

Quadratic differential systems, non-associative algebra,singular points, stability

Hrčak ID:

289606

URI

https://hrcak.srce.hr/289606

Publication date:

30.12.2022.

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