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.

Uniqueness of the Sum of Points of the Period-Five Cycle of Quadratic Polynomials

“…After this we concentrate on the period five case and show that the sum of period five cycle points is at most three-valued. The next result shows the relation between the sums of cycle points of the (x, y)-plane [7] and the (u, v)-plane [9]:…”

“…Based on the article [9], the equations of periodic orbits of period three and four are P 3 (u, v) = 0 and P 4 (u, v) = 0, where…”

“…Proof. By the article [9], the equation for period five orbit on the (u, v)-plane is of the form P 5 (u, v) = 0, where…”

“…The dynamics of quadratic polynomials is commonly studied by using the family of maps f c (x) = x 2 + c, where c ∈ C and x i+1 = f c (x i ) = x 2 i + c. In the article [9] we presented the corresponding iterating system on a new coordinate plane using the change of variables to the (x, y)-plane model (see [7]). In this new (u, v)-plane model, equations of periodic curves are of remarkably lower degree than in earlier models.…”