Аннотация:
This paper describes a predictive control method to
search for unstable periodic orbits of the generalized tent map.
The invariant set containing periodic orbits is a repelling set with
a complicated Cantor-like structure. Therefore, a simple local stabilization of the orbit may not be enough to find a periodic orbit,
due to the small measure of the basin of attraction. It is shown
that for certain values of the control parameter, both the local behavior and the global behavior of solutions change in the controlled
system; in particular, the invariant set enlarges to become an interval or the entire real axis. The computational particularities of
using the control system are considered, and necessary conditions
for the orbit to be periodic are given. The question of local asymp-
totic stability of subcycles of the controlled system’s stable cycles
is fully investigated, and some statistical properties of the subset
of the classical Cantor middle thirds set that is determined by the
periodic points of the generalized tent map are described.
