Chaos and GraphicsM and J sets from Newton's transformation of the transcendental mapping F(z)=ezw+c with vcps
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, , is given by . 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, with parameter c, the dynamical characteristics of fc(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)=ezw+c, with parameter w:
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 in the dynamical plane, which is defined in Section 3.1. As fw(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 fw(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 fw(z).
Section snippets
Iterating formula
Let z=x+iy and w=α+iβ, where x,y,α,β∈R. Let denote the modulus of z and θz∈(−π,π] be the principal argument of z in the dynamical plane. Let zn be the nth iteration of fw(z). It follows from (1) that
Critical point
A critical point of Newton's transformation fw(z) satisfies the condition that f′w(z)=0. Sincea critical point of fw(z) is
vcps
For a given value of w, it follows from (4) that the number of critical points is countably infinite. If we impose the condition that −π<arg(zk)⩽π, then the number of valid critical points becomes finite and it varies with the parameter w. Definition 1 For a given parameter w, the is composed of the critical points zk that satisfy the condition −π<arg(zk)⩽π, i.e.,
The range of k can be determined from (4) asIf vx≠0, we let
Filled-in Julia sets of fw(z)
Elegant filled-in Julia sets can be constructed by choosing appropriate w-values in the generalized M set defined in Section 3 and generated by the typical escaping-time algorithm, see Fig. 11. In Fig. 8, Fig. 9, Fig. 10, the filled-in Julia sets are generated by means of the convergence-time algorithm. The color is determined from the number of iterations that the orbit of a point in the dynamical plane is attracted to an orbit of a valid critical point. The procedure is described below:
- 1.
Choose
Conclusion
Newton's transformation fw(z)=z−1/(wzw−1) containing only one complex parameter w () is constructed from the transcendental mapping F(z)=ezw+c. For a given w, the number of the critical points of fw(z) is countably infinite. The generalized Mandelbrot set in the parameter plane is constructed from the Valid Critical Point Set (). For a given parameter w, if the orbits of all the valid critical points are attracted to either an n-cycle or a neutral fixed point, then w belongs to the
Acknowledgements
This research is supported by the National Natural Science Foundation of China, Subsidiary plan for core teachers of institutions of higher learning from the Ministry of Education of China, Liaoning Provincial Natural Science Foundation of China and Education Department Foundation of Liaoning Province of China.
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