  Periodic orbits are fundamental to understand the dynamics of nonlinear
systems. In this work, we focus on two aspects of interest regarding periodic
orbits, in the context of a dissipative mapping, derived from a prototype model
of a non-linear driven oscillator with fast relaxation and a limit cycle. For
this map, we show numerically the exponential growth of periodic orbits
quantity in certain regions of the parameter space and provide an analytical
bound for such growth rate, by making use of the transition matrix associated
with a given periodic orbit. Furthermore, we give numerical evidence to support
that optimal orbits, those that maximize time averages, are often unstable
periodic orbits with low period, by numerically comparing their performance
under a family of sinusoidal functions.
