On the Dynamics of a Third Order Newton's Approximation Method
Aurelian Gheondea, Mehmet Emre Şamcı
TL;DR
This paper analyzes the dynamical system generated by the third-order Newton-like iteration $M_f(x)=N_f(x)-\frac{f(N_f(x))}{f'(x)}$, where $N_f(x)=x-\frac{f(x)}{f'(x)}$, for Newton-type functions $f$ with at least four real roots and opposite-end behavior. The main result proves that $M_f$ has periodic points of every prime period and that the set of initial points with non-convergent orbits is uncountable, with damping and differentiability preserving these chaotic features; the Scaling Theorem shows the dynamics are invariant under affine conjugacy and extends to $M_{\lambda,f}$. The methodology combines interval-mapping lemmas, Sharkovsky-type constructions, and a reduction via affine conjugacy to simpler polynomial models. The results deepen the understanding of higher-order Newton schemes and provide a rigorous basis for observed chaotic behavior in $M_f$.
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.