Dynamics on ₵ with generalized Newton-Raphson maps: Julia sets, uncertainty dimension, and periodic orbits

Christian, JM; Jensen, GS

Abstract

The Newton-Raphson (NR) method is a well-known iterative scheme for approximating the roots of
functions. Deployed on the complex plane, ₵, perhaps its most famous application is to finding the
cube roots of –1. One often regards any specific outcome of the root-finding exercise as a secondary
consideration, and instead interprets the NR map as a two-dimensional discrete process with nonlinear
feedback. In so-doing, an extremely rich spectrum of dynamical phenomena emerges that includes
fixed points, periodic points, instabilities, unpredictability, fractal basin boundaries, and the Wada
property. Beyond their abstract nature, these concepts underpin modern understandings of physics.
Here, we investigate a direct extension of the standard NR class of maps. The procedure generates a
variant that is cubically convergent, but where the computational overhead involves additional
function evaluations at each iteration (including an awkward square-root operation). Binomial
expansions within that generalized formulation lead to a hierarchy of nonlinear maps on ₵, the lowest
order of which recovers the standard-NR algorithm. Each successively higher-order map can be
regarded as a dynamical system in its own right; for example, the next order corresponds to the
Schröder algorithm [A. S. Househölder, Principles of Numerical Analysis (Dover, New York, 1974)].
It is this hierarchy and the interconnected properties of its constituent maps that are of interest to us.
A systematic survey of results will be presented for a range of the lowest-order maps, with particular
emphasis placed on problems involving the N
th roots of –1. New families of fixed points have been
found in addition to the expected N roots, and the basins of attraction computed for each map.
Furthermore, the uncertainty dimension of the basin boundaries (which are examples of Julia sets)
have been estimated and parametrized by system properties. The inclusion of a complex relaxation
parameter has facilitated a linear stability analysis of the fixed points. That parameter leaves the fixed
points unchanged, but allows the fractal boundaries to be transformed (via stretching and twisting)
into patterns with enhanced complexity. Finally, periodic solutions have been identified by
deriving––and subsequently solving (with the aid of symbolic computation)––the polynomial
equations that prescribe period-2 and period-3 orbits. Accordingly, new formulae may be deduced for
predicting the number of period-M points in a given map (where M = 1, 2, 3, …). These orbits are
demonstrably unstable both analytically and numerically, and they are contained within the Julia set
of the corresponding map.

Citation

Christian, J., & Jensen, G. (2023, April). Dynamics on ₵ with generalized Newton-Raphson maps: Julia sets, uncertainty dimension, and periodic orbits. Poster presented at 64th British Applied Mathematics Colloquium (BAMC 2023), University of the West of England and University of Bristol, UK

• Presentation Conference Type: Poster
• Conference Name: 64th British Applied Mathematics Colloquium (BAMC 2023)
• Conference Location: University of the West of England and University of Bristol, UK
• Start Date: Apr 3, 2023
• End Date: Apr 5, 2023
• Acceptance Date: Feb 23, 2023
• Publication Date: Apr 5, 2023
• Deposit Date: Apr 11, 2023
• Publicly Available Date: May 18, 2023
• Publisher URL: https://rise.articulate.com/share/m_8PV5egFfp51rbRhApK6GtC3ZkRGmpN#/
• Related Public URLs: https://www.uwe.ac.uk/events/british-applied-mathematics-colloquium-bristol-2023
• Additional Information: Event Type : Conference