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This is a general question about Archimedian spirals.

I have the parameterized spiral: $$ \vec{u(t)}=(at+b)\hat{r} + (ct+d)\hat{\theta} $$

I have been leaving b=0 and d=0, so I am just worrying with the a and c constants. I am going to show a few combinations of these coefficients and the spirals they make to illustrate my question.

$$a=16 , c=74$$ a=16 , c=74 $$a=164 , c=740$$ enter image description here

So what is going on here causing the spiral to 'implode' or break down. I tried a lot of combinations of coefficients, but I can not seem to pin down the pattern on my own.

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    It seems theta is increasing much much faster than r can in a given step so the points become disconnected and widely spread apart, perhaps if you make the parameters closer you can recover a spiral of that's the case? – 2017-01-20

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This isn't anything to do with the spiral itself - this is a phenomenon commonly called "aliasing", and it is just a flaw in the program you're using. If you look at the first image, you'll notice that there are flat sides to it, meeting at sharp angles; but your formula involves only "smooth" functions, so there shouldn't be any corners. What's going on is that your graphing software is drawing the picture by evaluating $u(t)$ at various values of $t$, separated by a fixed step size, and just drawing straight lines in between these points. As long as the curve moves slowly, this is fine - you won't notice flat parts that are a millimeter long, for example. But once it starts moving quickly, the step size it's using starts skipping important parts of the curve, and you get parts that look flatter than they should. In your second picture, it looks like the curve is now moving so fast that in between two points that the software plots, the curve has already gone nearly halfway around. But the computer isn't clever enough to notice this, so it just draws a line that cuts directly from one point to the next, ignoring the fact that the actual spiral moves along a circle (almost) along the way. Importantly, the bigger you make $a$ and $c$ the faster the curve moves, so the more dramatic this effect will get.

Your software might have a setting for this; an option like "step size" would be the appropriate one to mess with. Try making it smaller (anything bigger than zero). You should start seeing smoother curves.