Changing Mandelbrot Formula

[hoddie54]

Using a custom fractal program I have changed the formula from Z_n = Z_n-1² + C to Z_n = Z_n-1^K + C, where both C and K are constants (K is real and an integer). So for example the formula could be Z_n = Z_n-1⁴ + C, and then it renders it as a Mandelbrot.

My question is a possible explanation for something very strange that I've encountered and I'm wondering if it is just a programming error or something in the fractal itself. Upon increasing the value of K to something like 12, a strange circle appears. Similarly if I decrease the zoom on the normal Mandelbrot (around 10^-38), I also get the circles.

Here are some pictures - I'll make more sense:[spoiler]
Normal Equations:

Zoom in on this circle (Normal Equation):

Where K = 10:

Where K = 10, and it is zoomed in onto the circle:
[/spoiler]

[lkmitch]

I'm guessing programming error, possibly related to the bailout.  Or maybe an overflow.  The nine-lobed critter for k=10 looks correct, though.