  Quantum states of the electromagnetic field are of considerable importance,
finding potential application in various areas of physics, as diverse as solid
state physics, quantum communication and cosmology. In this paper we introduce
the concept of truncated states obtained via iterative processes (TSI) and
study its statistical features, making an analogy with dynamical systems theory
(DST). As a specific example, we have studied TSI for the doubling and the
logistic functions, which are standard functions in studying chaos. TSI for
both the doubling and logistic functions exhibit certain similar patterns when
their statistical features are compared from the point of view of DST. A
general method to engineer TSI in the running-wave domain is employed, which
includes the errors due to the nonidealities of detectors and photocounts.
