  We extend and improve the existing characterization of the dynamics of
general quadratic real polynomial maps with coefficients that depend on a
single parameter $\lambda$, and generalize this characterization to cubic real
polynomial maps, in a consistent theory that is further generalized to real
$m$-th degree real polynomial maps. In essence, we give conditions for the
stability of the fixed points of any real polynomial map with real fixed
points. In order to do this, we have introduced the concept of Canonical
Polynomial Maps which are topologically conjugate to any polynomial map of the
same degree with real fixed points. The stability of the fixed points of
canonical polynomial maps has been found to depend solely on a special function
termed Product Position Function for a given fixed point. The values of this
product position determine the stability of the fixed point in question, when
it bifurcates, and even when chaos arises, as it passes through what we have
termed stability bands. The exact boundary values of these stability bands are
yet to be calculated for regions of type greater than one for polynomials of
degree higher than three.
