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

TL;DR

The paper introduces a new coordinate transformation for quadratic polynomials that simplifies the analysis of periodic orbits, reducing complex high-degree equations to quadratic ones in the transformed plane.

Contribution

A novel (u,v)-coordinate model is proposed, making it easier to analyze periodic orbits of quadratic polynomials by simplifying the equations involved.

Findings

Equation for period four orbits becomes quadratic in (u,v)-plane.

Transformation reduces complexity from degree 12 to degree 2.

Explicit solutions are possible in the new coordinate system.

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.