If $gcd(e,\lambda(N))=1,$ the sequence is purely periodic. Otherwise it may have an initial segment followed by a cycle, as you observe. Its maximal period divides the Carmichael function $\lambda(N)$ which is $\textrm{lcm}(p-1,q-1)$ when $N=pq,$ with $p,q$ prime.

Note that $2$ divides $\lambda(N)$ in this case, and explains the tendency in the period which you noticed. The paper *Period of the power generator and small values of the Carmichael function* by Friedlander, Pomerance and Shparlinski 
available [here][1]
has a detailed discussion.


  [1]: http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-00-01282-5/S0025-5718-00-01282-5.pdf