Existence of a cylinder foliated by periodic orbits in the generalized Chazy differential equation

TL;DR

This paper proves the existence of invariant cylinders foliated by periodic orbits in a generalized Chazy differential equation, extending previous numerical observations and providing an algorithm to verify conditions for such structures for various parameter values.

Contribution

The paper analytically establishes the existence of periodic solutions in the generalized Chazy equation and introduces an algorithm to verify invariant cylinders for any positive integer parameter.

Findings

Existence of periodic solutions for specific parameter values.

An algorithm to check conditions for invariant cylinders.

Successful application of the algorithm up to q=100.

Abstract

The generalized Chazy differential equation corresponds to the following two-parameter family of differential equations \begin{equation*}\label{gcdeq} \dddot x+|x|^q \ddot x+\dfrac{k |x|^q}{x}\dot x^2=0, \end{equation*} which has its regularity varying with $q$ , a positive integer. Indeed, for $q=1$ it is discontinuous on the straight line $x=0$, whereas for $q$ a positive even integer it is polynomial, and for $q>1$ a positive odd integer it is continuous but not differentiable on the straight line $x=0$. In 1999, the existence of periodic solutions in the generalized Chazy differential equation was numerically observed for $q=2$ and $k=3$. In this paper, we prove analytically the existence of such periodic solutions. Our strategy allows to establish sufficient conditions ensuring that the generalized Chazy differential equation, for $k=q+1$ and any positive integer $q$ , has actually an invariant topological cylinder foliated by periodic solutions in the $(x,\dot x,\ddot x)$-space. In order to set forth the bases of our approach, we start by considering $q=1,2,3$, which are representatives of the different classes of regularity. For an arbitrary positive integer $q$, an algorithm is provided for checking the sufficient conditions for the existence of such an invariant cylinder, which we conjecture that always exists. The algorithm was successfully applied up to $q=100$.