Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 43, 2018, 859-883

PERIODIC ORBITS 1–5 OF QUADRATIC POLYNOMIALS ON A NEW COORDINATE PLANE

Pekka Kosunen

University of Eastern Finland, Department of Physics and Mathematics
P.O. Box 111, FI-80101 Joensuu, Finland; pekka.kosunen 'at' uef.fi

Abstract. While iterating the quadratic polynomial f_c(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.

2010 Mathematics Subject Classification: Primary 37F10; Secondary 30D05.

Key words: Periodic orbit, iteration, quadratic polynomial, eigenvalue, bifurcation diagram.

Reference to this article: P. Kosunen: Periodic orbits 1–5 of quadratic polynomials on a new coordinate plane. Ann. Acad. Sci. Fenn. Math. 43 (2018), 859-883.

Full document as PDF file

https://doi.org/10.5186/aasfm.2018.4356