Autonomous Planar Systems of Riccati Type

Autonomous Planar Systems of Riccati Type

Year:    2022

Author:    Gary R. Nicklason

Journal of Nonlinear Modeling and Analysis, Vol. 4 (2022), Iss. 2 : pp. 171–197

Abstract

The role of Riccati type systems in the plane along with the related linear, second order differential equation is examined. If $x$ and $y$ are the variables of the Riccati differential equation, then any integrable Riccati system has two independent invariant curves dependent upon these variables whose nature is easily determined from the solution of the linear equation. Each of these curves has the same cofactor. Other invariant curves depend upon $x$ alone and are shown to be less important. The systems have both Liouvillian and non–Liouvillian solutions and are easily transformable to symmetric systems. However, systems derived from them may not be symmetric in 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 Abel differential equations.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.12150/jnma.2022.171

Journal of Nonlinear Modeling and Analysis, Vol. 4 (2022), Iss. 2 : pp. 171–197

Published online:    2022-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    27

Keywords:    Riccati differential equation Centre–focus problem Algebraic invariant curve Cofactor Symmetric centres.

Author Details

Gary R. Nicklason