Glasnik matematički, Vol. 55 No. 2, 2020.
Original scientific paper
https://doi.org/10.3336/gm.55.2.11
Partial qualitative analysis of planar \(\mathcal{A}_{Q}\)-Riccati equations
Borut Zalar
; University of Maribor, Faculty of civil engineering, Transportation engineering and architecture, Smetanova 17, 2000 Maribor, Slovenia
Brigita Ferčec
; University of Maribor, Faculty of energy technology, Hočevarjev trg 1, 8270 Krško, Slovenia
Yilei Tang
; School of mathematical sciences, Shanghai Jiao Tong University, 800 Dongchuan Road, Minhang District Shanghai, 200240, China
Matej Mencinger
; University of Maribor, Faculty of civil engineering, transportation engineering and architecture, Smetanova 17, 2000 Maribor, Slovenia
Abstract
If we view the field of complex numbers as a 2-dimensional commutative real algebra, we can consider the differential equation \(z^{\prime}=az^{2}+bz+c\) as a particular case of $\mathcal{A}-$ Riccati equations $z^{\prime} =a\cdot(z\cdot z)+b\cdot z+c$ where $\mathcal{A=(}\mathbb{R}^{n},\cdot)$ is a commutative, possibly nonassociative algebra, $a,b,c\in\mathcal{A}$ and $z:I\rightarrow\mathcal{A}$ is defined on some nontrivial real interval. In the case $\mathcal{A}=\mathbb{C}$, the nature of (at most two) critical points can be described using purely algebraic conditions involving involution $\ast$ of $\mathbb{C}$. In the present paper we study the critical points of $\mathcal{L}(\pi)-$ Riccati equations, where $\mathcal{L}(\pi)$ 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 $\infty,$ 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 $\mathcal{L}(\theta)$.
Keywords
Differential systems; Riccati equation; commutative algebra; singular points; stability; center problem
Hrčak ID:
248671
URI
Publication date:
23.12.2020.
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