  Self-similar stable mixed moving average processes can be related to
nonsingular flows through their minimal representations. Self-similar stable
mixed moving averages related to dissipative flows have been studied, as well
as processes associated with identity flows which are the simplest type of
conservative flows. The focus here is on self-similar stable mixed moving
averages related to periodic and cyclic flows. Periodic flows are conservative
flows such that each point in the space comes back to its initial position in
finite time, either positive or null. The flow is cyclic if the return time is
positive.
  Self-similar mixed moving averages are called periodic, respectively cyclic,
fractional stable motions if their minimal representations are generated by
periodic, respectively cyclic, flows. These processes, however, are often
defined by a nonminimal representation. We provide a way to detect whether they
are periodic or cyclic even if their representation is nonminimal. By using
these identification results, we obtain a more refined decomposition of
self-similar mixed moving averages.
