Periodic orbits 1-5 of quadratic polynomials on a new coordinate plane

Abstract

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.