Periodic solutions and associated limit cycle for the generalised Chazy equation

We study the generalised Chazy equation, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\dddot x + x^q \ddot x + kx^{q - 1} \dot x^2 = 0$$ \end{document}, which is characterised by the symmetries of time translation and rescaling. For a large class of initial conditions numerical computations reveal the asymptotic appearence of periodic solutions for k=q+1. These solutions are identical after rescaling and, in this sense, exhibit the property of a limit cycle in the three dimensional phase space. The periodic solutions are related to a conventional limit cycle of a class of second order ordinary differential equations which are connected to the existence of a first integral of the generalised Chazy equation.