Abstract
This paper studies the Kummer–Schwarz differential equation \({2 \dot{x}\, \dddot{x}- 3 \ddot x^2=0}\) which is of special interest due to its relationship with the Schwarzian derivative. This differential equation is transformed into a first order differential system in \({\mathbb{R}^3}\) , and we provide a complete description of its global dynamics adding the infinity.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Abraham-Shrauner B., Leach P., Govinder K., Ratcliff G.: Hidden and contact symmetries of ordinary differential equations. J. Phys. A: Math. Gen. 28, 6707–6716 (1995)
Cima A., Llibre J.: Bounded polynomial vector fields. Trans. Amer. Math. Soc. 318, 557–579 (1990)
Dumortier F.; Llibre J.; Artés J.C.: Qualitative theory of planar differential systems, UniversiText, Springer–Verlag, New York, 2006
Goviender K., Leach P.: The algebraic structure of the first integrals of third-order linear equations. J. Math. Anal. Appl. 195, 114–133 (1995)
Leach P.: On the uniqueness of the Schwarzian and linearisation by nonlocal contact transformation. J. Math. Anal. Appl. 235, 84–107 (1999)
Leach P.: Symmetry and singularity properties of the generalised Kummer-Schwarz and related topics. J. Math. Anal. Appl. 348, 487–493 (2008)
Vidal C., Gómez P.: An extension of the Poincaré compactification and a geometric interpretation. Proyecciones 22(3), 161–180 (2003)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Llibre, J., Vidal, C. Global Dynamics of the Kummer–Schwarz Differential Equation. Mediterr. J. Math. 11, 477–486 (2014). https://doi.org/10.1007/s00009-013-0299-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00009-013-0299-4