### Borut Zalar, Brigita Ferčec, Yilei Tang and Matej Mencinger

University of Maribor, Faculty of civil engineering, Transportation engineering and architecture, Smetanova 17, 2000 Maribor, Slovenia
e-mail: borut.zalar@um.si

University of Maribor, Faculty of energy technology, Hočevarjev trg 1, 8270 Krško, Slovenia
and
Center for applied mathematics and theoretical physics, University of Maribor, Mladinska 3, 2000 Maribor, Slovenia
e-mail: brigita.fercec@um.si

School of mathematical sciences, Shanghai Jiao Tong University, 800 Dongchuan Road, Minhang District Shanghai, 200240, China
e-mail: mathtyl@sjtu.edu.cn

University of Maribor, Faculty of civil engineering, transportation engineering and architecture, Smetanova 17, 2000 Maribor, Slovenia
and
Institute of mathematics, physics and mechanics, Jadranska 19, 1000 Ljubljana, Slovenia
and
Center for applied mathematics and theoretical physics, University of Maribor, Mladinska 3, 2000 Maribor, Slovenia
e-mail: matej.mencinger@um.si

Abstract.   If we view the field of complex numbers as a 2-dimensional commutative real algebra, we can consider the differential equation z'=az2+bz+c as a particular case of 𝓐- Riccati equations z'=a · (z · z)+b · z+c where 𝓐=( ℝn,·) is a commutative, possibly nonassociative algebra, a,b,c∈𝓐 and z:I → 𝓐 is defined on some nontrivial real interval. In the case 𝓐=ℂ, the nature of (at most two) critical points can be described using purely algebraic conditions involving involution * of . In the present paper we study the critical points of 𝓛(π)- Riccati equations, where 𝓛(π) is the limit case of the so-called family of planar Lyapunov algebras, which characterize 2-dimensional homogeneous systems of quadratic ODEs with stable origin. The number of possible critical points is 1, 3 or ∞, depending on coefficients. The nature of critical points is also completely described. Finally, simultaneous stability of the origin is considered for homogeneous quadratic part corresponding to algebras 𝓛(θ).

2010 Mathematics Subject Classification.   34A34, 34C60, 17A99

Key words and phrases.   Differential systems, Riccati equation, commutative algebra, singular points, stability, center problem

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