On the Dynamics of a Third Order Newton's Approximation Method

TL;DR

This paper investigates the complex dynamics of a third order Newton's approximation method, revealing the existence of periodic points of all prime periods and uncountably many initial points with non-converging sequences, and extends some theoretical results.

Contribution

It demonstrates that the third order approximation function exhibits rich dynamical behavior, including periodic points of all prime periods, and generalizes the Scaling Theorem to differentiability.

Findings

Existence of periodic points of any prime period.

Uncountably many initial points lead to non-convergent sequences.

The Scaling Theorem applies under differentiability, not just analyticity.

Abstract

We show that the third order approximation function $M_f$, proposed by S. Amat, S. Busquier, S. Plaza, in \textit{J. Math. Anal. Appl.}, 366(2010), 24--32, for functions $f$ twice continuously differentiable and such that both $f$ and its derivative do not have multiple roots, with at least four roots, and infinite limits of opposite signs at $\pm\infty$, have periodic points of any prime period and that the set of points $a$ at which the approximation sequence $(M_f^n(a))_{n\in\mathbb{N}}$ does not converge is uncountable. In addition, we observe that in their Scaling Theorem analyticity can be replaced with differentiability.