On the Differential Equation $\frac{d}{dt}\left(\frac{\cos x}{1-\dot{x}}\right)\,=\,-\sin x$

TL;DR

This paper analyzes a singular autonomous differential equation from a dynamical systems perspective, describing all phase plane orbits, revealing complex behaviors like infinite critical points, periodic orbits, and non-disjoint trajectories due to singularities.

Contribution

It provides a detailed phase plane analysis of a singular differential equation, including explicit solutions and orbit classification, which was not previously explored.

Findings

The equation has infinitely many equilibrium points.

Periodic orbits surround each equilibrium point.

Non-disjoint orbits occur in certain phase plane regions.

Abstract

The autonomous differential equation in the title is derived in S.Srinivasan~\cite{ss05} (equation (E) in ~\cite{ss05}) in the context of certain discrete sums from the number theoretic considerations. These discrete sums are then estimated in terms of an integral involving the solutions of this differential equation; no analysis is done on this integral in ~\cite{ss05}. Our main objective is to consider this equation from the dynamical systems view point and describe all the orbits in the phase plane. The equation is singular in the sense that the coefficient of $\ddot{x}$ vanishes or becomes unbounded at a few points. This singularity puts a restriction on the initial data. The solutions that we obtain in explicit form are, however, smooth and satisfy the equation pointwise or in the limiting sense. The explicit form of solutions may also be used to analyse the integral in ~\cite{ss05}. The equation possesses an infinite number of equilibrium or critical points and there are periodic orbits surrounding each of them in a part of the phase plane. In the complement of this part of the phase plane, there are non-periodic orbits, any two of which have an infinite number of common points. This, however, is in great contrast with the fact that any two (distinct) orbits of a regular autonomous system are disjoint, thus bringing out the singular nature of the equation