Matična publikacijaGlasnik matematički (Online)
Način izrade datotekeizvorno digitalna građa
OpisIf 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).