Glasnik Matematicki, Vol. 57, No. 2 (2022), 251-264. \( \)

ON A GENERALIZATION OF SOME INSTABILITY RESULTS FOR RICCATI EQUATIONS VIA NONASSOCIATIVE ALGEBRAS

Hamza Boujemaa and Brigita Ferčec

Département de Mathématiques, Faculté des Sciences de Rabat, Mohammed V University in Rabat, 1014RP Rabat, Rabat, Morocco
e-mail:hamzaboujemaa@gmail.com

Faculty of Energy Technology, University of Maribor, Hočevarjev trg 1, 8270 Krško, Slovenia, &, Center for Applied Mathematics and Theoretical Physics, University of Maribor, Mladinska 3, SI-2000 Maribor, Slovenia, &, Faculty of natural sciences and mathematics, University of Maribor, Koroška c. 160, 2000 Maribor, Slovenia
e-mail:brigita.fercec@um.si

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.

2020 Mathematics Subject Classification.   34A34, 17A99

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

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

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