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

Abstract

While iterating the quadratic polynomial fc(x) = x2 + 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 the situation drastically. The parametrization in the new model is simpler in the sense that, in it, the equations of the periodic orbits are of lower degree than the ones in previous models. 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 ((((x2 + c)2 + c)2 + c)2 + c – x)/((x2 + c)2 + c – x) = 0 is of degree 12 and it is thus much more difficult to solve.