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.