M and J sets from Newton's transformation of the transcendental mapping F(z)=e^zw+c with vcps

Abstract

Newton's transformation f_w(z)=z−1/(wz^w−1) containing only one complex parameter w (w≠0 or 1) is constructed from the transcendental mapping F(z)=e^zw+c. Although the number of critical points of f_w(z) is countably infinite, a method based on the Valid Critical Point Set, $\text{vcps}\text{={}\text{z}{}_{\mathrm{k}}\text{∈C}\text{−π<}\text{arg}\text{(z}{}_{\mathrm{k}}\text{)⩽π,}\text{f′}{}_{\mathrm{w}}\text{(z}{}_{\mathrm{k}}\text{)=0,}\text{k∈Z}}$, is discussed for the generation of the generalized Mandelbrot set M of f_w(z). The petal fragments, the multi-period fragments, and the classical “Mandelbrot set” fragments are found in M. The dynamical characteristics of f_w(z) for different values of w are analyzed. The relationship between the parameter w in a classical “Mandelbrot set” fragment and the structure of the corresponding filled-in Julia set is investigated. A large number of novel images of filled-in Julia sets are generated based on the rate of convergence of orbits.

Introduction

In the research of visual presentation of nonlinear dynamical systems using the computer, exotic Julia images were generated by means of nonlinear mappings such as polynomial, transcendental and equivariant mappings [1], [2], [3], [4], [5], [6]. For a family of mappings governed by a parameter, the image structure of a filled-in Julia set may change abruptly as the parameter varies.

In numerical analysis, Newton's method, the secant method and other modified methods are powerful tools for finding solutions of a nonlinear equation or a system of nonlinear equations. If these methods are applied to the investigation of visual presentation of nonlinear dynamical systems, new models of filled-in Julia sets can be obtained [7], [8], [9], [10]. Newton's transformation of a nonlinear equation, $\text{F(z)=0}\text{(z∈C)}$, is given by $\text{f(z)=z−F(z)}\text{F′(z)}$. In [11], Newton's method was applied to solve the cubic equation F(z)≡z³+(c−1)z—c=0, where c is a complex parameter. For c=1, the basins of attraction of the roots can be found in many popular books such as [1] and their boundaries possess fractal characteristics. In general, the filled-in Julia set of Newton's transformation f(z) is composed of the union of the attracting basins of attracting fixed points. However, if f(z) is extended to a family of mappings, $\text{f}{}_{\mathrm{c}}\text{(z)}\text{(c∈C)}$ with parameter c, the dynamical characteristics of f_c(z) may be significantly different from that of the original transformation, and the relative filled-in Julia sets may be composed of the union of attracting basins of orbits with periods higher than 1. Now, the question is: how can we find those values of c with such a property? For the familiar mapping, f(z)=z²+c, it is well known that the c-values generating Julia sets with various structures can be obtained from the Mandelbrot set in the parameter plane. Similarly, for a family of Newton's transformations, we may also construct the generalized M set which provides the c-values for the generation of filled-in Julia sets with different structures.

The dynamics of transcendental functions are quite different from that of polynomial functions, mainly because of essential singularities [12]. In this paper, we consider the family of Newton's transformations of the transcendental mapping, F(z)=e^zw+c, with parameter w:

$$\text{f}{}_{\mathrm{w}}\text{(z)=z−}\text{F(z)}\text{F′(z)}\text{=z−}\text{1}\text{wz}{}^{\mathrm{w-1}}\text{(z,w∈C,}\text{w≠0}\text{or}\text{1).}$$

A critical point of f(z) is defined by f′(z)=0. Since the critical points of (1) are countably infinite, we construct the generalized M set in the parameter plane by considering only the Valid Critical Point Set $\text{(}\text{vcps}\text{)}$ in the dynamical plane, which is defined in Section 3.1. As f_w(z) contains neutral points, the construction of the generalized M set is somewhat different from that of the classical Mandelbrot set. Its definition is given in Section 3.2. In Section 2, we will derive some basic properties of f_w(z). The constructions of the generalized M set and filled-in Julia sets, and a discussion of their dynamical characteristics are given in 3 Generalized, 4 Filled-in Julia sets of, respectively. A large number of exotic filled-in Julia sets can be obtained from f_w(z).