We are not sure whether MD5 has fixed point or not. But since the sample space is finite, it must have cycles:
$$ A →(MD5)→ B →(MD5)→ C →(MD5)→ D →(MD5)→ A $$
Has any research been done on MD5 to find cycles?
What caused I think about it is that if S is the sample space and R1 the range of MD5(once or taken in any number) them R1⊆S also R2⊆R1
MD5(one or multiple): S → R1
MD5(one or multiple): R1 → R2
...
Not yet*.
The only major project to try to find a cycle shut down in 2004 once a general collision vulnerability was found, due to the fact that a latter is a much more severe vulnerability.
I was unable to find anything other than that Wikipedia article and this very thread on the topic. (There is a group that looked into ways to improve the internal state's cycle length, to help engineer future editions or hash functions to be more robust; however, this is unrelated to cycles of the digest itself when feeding the digest back into the function wholesale.)
*Do you want to find one?
To get a cycle for MD5 would require nowhere near as much power as breaking it.
Wikipedian Taxman estimated in 2004 that it would take a 12.25-Teraflops supercomputing cluster about 1.77Ms (i.e. just under 3 weeks) to find a cycle. These days, an arbitrary provider, Nimbix, could allegedly provide this [if my math is correct] on the order of 1 business day and $800.