Goodwin oscillator explained
The ODEs for the Goodwin oscillator are: 

$$\begin{align}
\frac{dX}{dt}=&amp;\ k_1\frac{K_i^n}{K_i^n+Z^n} -k_2X\\[1em]
\frac{dY}{dt}=&amp;\ k_3X-k_4Y\\[1em]
\frac{dZ}{dt}=&amp;\ k_5Y-k_6Z\end{align}$$

If you assume $X$ to be mRNA and $Y$to be protein then $Z$ can be thought of as an activated form of the protein (lets for instance assume a phosphorylated form).

The Hill coefficient, $n$ roughly tells you how many monomers of the activated protein co-operatively bind to the promoter or the number of protein monomers that form a multimeric repressor complex which inhibits the production of $X$. $K_i$ is a measure of how strongly would $X$ be inhibited by $Z$. To be precise it is equal to the concentration of $Z$ at which the production of $X$ is half its maximum rate. (Similar to Michaelis constant)

What the model says is that at higher values of $n$ (>8) there are sustained oscillations (which are called limit cycles). At lower values the oscillations damp. Note that even for low values of $n$ sustained oscillations (but not limit cycle types) can arise if there is substantial delay in the loop. However these oscillations are not as stable as limit cycles to perturbations. 

Have a look at this post. 

These are some advanced concepts and you may start by reading this book by Steven Strogatz called Non-linear Dynamics and Chaos*.


*Strogatz, Steven H. Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. Westview press, 2014. (latest edition);  ISBN-13: 978-0813349107; ISBN-10: 0813349109