Autonomous Planar Systems of Riccati Type (Journal of Nonlinear Modeling and Analysis, 2024)
The role of Riccati type systems in the plane along with the related linear, second order differential equation is examined. If $x$ and $y$ arethe variables of the Riccati differential equation, then any integrable Riccatisystem has two independent invariant curves dependent upon these variableswhose nature is easily determined from the solution of the linear equation.Each of these curves has the same cofactor. Other invariant curves dependupon $x$ alone and are shown to be less important. The systems have bothLiouvillian and non–Liouvillian solutions and are easily transformable to symmetric systems. However, systems derived from them may not be symmetricin their transformed variables. Several systems from the literature are discussed with regard to the forms of the invariant curves presented in the paper.The relation of certain Riccati type systems is considered with respect to Abeldifferential equations.
Keywords: Riccati differential equation; Centre–focus problem; Algebraic invariant curve; Cofactor; Symmetric centres.