Partial qualitative analysis of planar 𝓐Q-Riccati equations

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 ℂ. 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).