Periodic solutions and associated limit cycle for the generalised Chazy equation

Abstract

We study the generalised Chazy equation, $\dddot x + x^q \ddot x + kx^{q - 1} \dot x^2 = 0$, 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.