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I have seen something similar happen before, but it usually happens when I have been on the computer for a very long time, and have had many applications running during that time.  Usually I just close everything and shutdown the computer (a complete power-off).  Then wait a minute before booting back up.  When I try it again, the "problem" no longer happens.     
 

Hi, sorry for the extremely late reply to this - I've had this thread open in a tab for ages meaning to answer!

The problem has two root causes: incorrect generation and use of random numbers in the iteration core, basically just faulty statistics. The other problem is unguarded memory access by multiple threads to the histogram/image buffer. Both problems get much worse with multi-threading in various ways; FarDareisMai on deviantArt has made a comprehensive study of these problems, but it's still not published anywhere unfortunately. I've looked at the Apophysis code now and then over the years (to try and figure out weird render differences with Chaotica), and as far as I can tell this problem is deeply entrenched in the old codebase, and with the lack of developers working on it I don't imagine it'll be fixed anytime soon :S

I finally answer this now because there was recently some discussion about a new GPU renderer, with questions about correctness in this regard: http://www.fractalforums.com/announcements-and-news/fractorium-a-new-gpu-accelerated-flam3-fractal-flame-renderer/

I also did some investigation of this stuff for JWildfire, and found out in my case that attempts of "doing it right" are not worth the effort, i. E. there was no visible difference in the result in the most cases (I measured this by calculating the image using different methods and compared the results using an algorithm, not just by "looking" )

But, the really big deal was the chosen random-number-generator (in means of quality).

I also use one instance of such a random-number-generater per render thread which works very fine.

My currently favourite one is an algorithm described by Marsaglia, but you can choose from different random-number-generators in JWildfire, probably the best is MersenneTwister, but the current implementation is far from perfection.

But in real-world-scenarios (movies, images posted as JPG, ...) you wont see much differences in the most cases.

Andreas, good to see you here!

Good info from both of you, thanks. What are your thoughts on ISAAC?