I'm reading an article which describes a generation of artificial AKG signal. The article states that they use the following equation of motion:
$ \left \lbrace \begin{array} {ccc} \dot x &=& \alpha x - \omega y \\ \dot y &=& \alpha y + \omega x\end{array}\right .$
where $\alpha = 1- \sqrt{x^2 + y^2}$ and $\omega$ is the angular velocity.
I have figured out the easy part where I use the $\theta$ and $r(=1)$ coordinates thus:
$ \left \lbrace \begin{array} {ccc} \dot x &=& \frac{d}{dt} \cos (\theta) = -\sin (\theta) \frac{d}{dt} = -\omega y \\ \dot y &=& \frac{d}{dt} sin (\theta) = \cos (\theta) \frac{d}{dt} = \omega x\end {array}\right .$
Which has made me assume that $r = r(t)$ and $\langle r\rangle = 1$. I have tried to solve this simple deferential equation:
$\frac{dr}{dt} = 1-r$
but the solution I have found was:
$r(t) = 1-c\text{e}^{t} $
Which has many problems, units to say the least.
It is clear to me that the units problem is originated from the definition of $\alpha$ but there is no reference to that in the article.
I was hoping somebody of you has encountered such a problem or is handling such approach.
Edit I have forgotten to mentioned the very important fact that they are using a unit circle. Also, my main issue is with the $\alpha$ part which, on the unit circle should be 0...
Edit A link to the article