Near-idempotents, near-nilpotents and stability of critical points for Riccati equations

Zalar, Borut; Mencinger, Matej

doi:10.3336/gm.53.2.06

Cited by 3 publications

(12 citation statements)

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“…After his classification of planar systems, many other authors considered various classifications, mostly limited to k = 2, see [4,5,6,7,8,18,22]. For systems (1.1), the question of (in)stability of singular point(s) is nontrivial, see [3,19,23,24,25,28], since the origin is a nonelementary nonhyperbolic singular point in any dimension and for any degree of homogeneity. There are also some papers about (algebraic) structure(s) and dynamics in systems (1.1), e.g.…”

Section: Introductionmentioning

confidence: 99%

“…In [28], the authors studied the stability of non-zero singular points of a quadratic system (1.1) using the so called λ−space of a nonzero element u defined by…”

Section: Introductionmentioning

confidence: 99%

“…In [3], the authors proved that the nilpotent line Rn consists of unstable singular points, if n is included in A λ (p) for some idempotent p ∈ A and that this result remains true even when p is not necessarily an idempotent but an element satisfying a weaker algebraic condition. In [28], the authors defined a more general algebraic framework in which the first main result of [3] was reinterpreted in sense of the following two definitions.…”

Section: Introductionmentioning

confidence: 99%

“…Definition 1.2 ( [28]). Let A be a commutative real algebra of dimension m and N k ⊂ A the subspace spanned by a set of linearly independent nilpotent elements n…”

Section: Introductionmentioning

confidence: 99%

“…

In [28], for any real non associative algebra of dimension m ≥ 2, having k linearly independent nilpotent elements n 1 , n 2 , . .

…”

mentioning

confidence: 99%

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

Boujemaa¹,

Ferčec²

2022

Glas. Mat. Ser. III

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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.

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Section: Introductionmentioning

confidence: 99%

“…In [28], the authors studied the stability of non-zero singular points of a quadratic system (1.1) using the so called λ−space of a nonzero element u defined by…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Definition 1.2 ( [28]). Let A be a commutative real algebra of dimension m and N k ⊂ A the subspace spanned by a set of linearly independent nilpotent elements n…”

Section: Introductionmentioning

confidence: 99%

“…

In [28], for any real non associative algebra of dimension m ≥ 2, having k linearly independent nilpotent elements n 1 , n 2 , . .

…”

mentioning

confidence: 99%

See 3 more Smart Citations