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.
It is well known that the sum of points of the period-five cycle of the quadratic polynomial ( ) = 2 + is generally not one-valued. In this paper we will show that the sum of cycle points of the curves of period five is at most three-valued on a new coordinate plane and that this result is essentially the best possible. The method of our proof relies on a implementing Gröbner-bases and especially extension theory from the theory of polynomial algebra.
The dynamics of quadratic polynomials is commonly studied by using the family of maps f c (x) = x 2 + c, where c ∈ C. In this paper we form equations of periodic orbits of periods six and seven on a new (u, v)-plane and consider also the corresponding equations on the (x, y)-plane. The new (u, v)-model produces equations for the periods six and seven with significantly lower degree than the ones obtained by the previous models which enables us to find their solutions.
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