I apologise for this simple question and I am sure what I am missing here is quite obvious. I completed a question earlier in which I had to rewrite the differential equation
$d^2\theta/dt^2+\delta d\theta/dt +sin(\theta)=x$
$d^2x/dt^2-x+x^3=0$
$(\theta,x)\in S^1\times \mathbb{R}^1$
as a first order system. I proceeded fine and the first order system I produced was correct except I took $d\theta/dt=\phi\in S^1$ where the provided solution instead took $d\theta/dt=v\in\mathbb{R}$. Why is this the case?
In case people would like to see I shall provide the first order system I produced here:
$d\phi/dt+\delta\phi+sin(\theta)=x$
$dy/dt-x+x^3=0$
$d\theta/dt=\phi, \theta \in S^1$ and $x,y \in\mathbb{R}$