  While iterating the quadratic polynomial f_{c}(x)=x^{2}+c the degree of the
iterates grows very rapidly, and therefore solving the equations corresponding
to periodic orbits becomes very difficult even for periodic orbits with a low
period. In this work we present a new iteration model by introducing a change
of variables into an (u,v)-plane, which changes situation drastically. As an
excellent example of this we can compare equations of orbits period four on
(x,c)- and (u,v)-planes. In the latter case, this equation is of degree two
with respect to u and it can be solved explicitly. In former case the
corresponding equation
((((x^{2}+c)^{2}+c)^{2}+c)^{2}+c-x)/((x^{2}+c)^{2}+c-x)=0 is of degree 12 and
it is thus much more difficult to solve.

Dynamical Systems