On the periodic behavior of the generalized Chazy differential equation

We consider the periodic behavior of the generalized Chazy differential equation x ⃛ + | x | q x ¨ + k | x | q x x ˙ 2 = 0 , where q is a positive integer and k is a real number. We give a pure analysis on the existence of non-trivial periodic solutions for k = q + 1 and the non-existence of them for k ≠ q + 1. Our method is based on considering the projections of the orbits onto the phase plane ( x , x ˙ ). We find that a non-trivial periodic solution of the equation is equivalent to a closed curve formed by two equilibrium points and two orbits with some specific constraints in the corresponding planar system and that the existence of such closed curves can be obtained by the existence of real zeros of some returning map. Our conclusion covers all q, which completes a recent result.