Given that a parametric eq for a circle is given by :
$$x= r \cos \theta \\ y= r \sin \theta $$
Is it possible to move the center of circle by a (periodic) function $f(r,\theta)$:
$$\begin{align} x &= r \cos \theta + f(r,\theta)\\ y &= r \sin \theta \end{align}$$ to obtain an ellipse?
Actually to define an ellipse in parametric equations, we just change the coefficients on the $\sin\theta$ and $\cos\theta$ terms.
$$x(t)=a\cos t$$ $$y(t)=b\sin t$$
So to answer your question, yes. $f(r,\theta)=n\cdot \cos\theta, n\in \mathbb R$. If we add any multiple of $r\cos$ to our $x$ part we obtain an ellipse.
$$x(t)=r\cos t+(n\cos t)\implies x(t)=(n+r)\cos t$$