Elsevier

Computers & Graphics

Volume 31, Issue 2, April 2007, Pages 301-307
Computers & Graphics

Generalized Binet dynamics

https://doi.org/10.1016/j.cag.2006.10.001Get rights and content

Abstract

The Binet formula provides a mechanism for the Fibonacci numbers to be viewed as a function of a complex variable. The Binet formula may be generalized by using other bases and multiplicative parameters that also give functions of a complex variable. Thus, filled-in Julia sets that exhibit escape time may be constructed. Moreover, these functions have computable critical points and hence we can create escape time images of the critical point based upon the underlying multiplicative parameter. Like the classic Mandelbrot set, these parameter space images provide an atlas of Julia sets.

Introduction

The Fibonacci numbers are given by the sequence 0, 1, 1, 2, 3, 5, 8, 13, … where each term is the sum of the previous two. This sequence can be defined via the recursive formulas: F0=0, F1=1, and Fn=Fn−1+Fn−2. However, we can also use the well-known Binet formula [1], [2], [3], which may be described as follows. Let τ=(1+5)/21.618 and τ¯=(1-5)/2-0.618 where τ is the golden ratio and τ¯ is the algebraic conjugate of τ. Then the Binet formula, F(z)=(τz-τ¯z)/5, is a complex function that is a generalization of the Fibonacci sequence. However, because this function involves the negative base τ¯, it takes complex values along the real line. Thus, while it gives a generalization to the complex domain, it does not give a generalization to the real domain.

In [4], the escape time of F(z) was examined, an interesting spiral around the origin was observed. This current investigation is the final step in the natural progression of studies that began with the observation of a spiral, which was followed by considering functions with a multiplicative parameter that made the spiral more dramatic. Those functions were then considered with a generalized base. When we realized that the critical points could be computed, we were ready to study the dynamics of the critical points. Thus, in this investigation, we study the dynamics of functions similar to the function F(z) that was studied in [4], but with two parameters of generalization. We are able, in a manner analogous to the classic Mandelbrot set, to use the critical point dynamics to locate Julia sets with visually rich behavior.

Specifically, we consider the complex dynamics of functions of the form fα,q(z)=f(z)=α(τz-τ¯z) where τ=(1+q)/2 andτ¯=(1-q)/2. While the function depends upon the parameters α and q, we will usually call the function f(z) for brevity. Of course, the Binet formula occurs when α=1/5 and q=5. We will determine critical points for these functions and create images of the behavior of the critical point relative to the parameter α. These critical point images are an analog of the classic Mandelbrot set [5], [6], [7], [8], [9], [10], and like the classic Mandelbrot set, we are able to use these images to identify values of α that give rise to complicated filled-in Julia sets for these families of functions. The function fα,q(z) is also related to the exponential functions αez; each of the terms of fα,q(z) corresponds to an exponential function. The dynamics of αez are discussed in [11], [12].

Section snippets

The critical points

Given f(z)=α(τz-τ¯z) we see that f(z)=α(τzln(τ)-τ¯zln(τ¯)). We note that when q>1, then τ¯<0 and hence ln(τ¯) is multi-valued. If we formally solve for the critical points; that is, solve f(z=)0 for z, we obtain z=ln(ln(τ¯)/ln(τ))/(ln(τ)-ln(τ¯)). Since the logarithm is multi-valued, there will be many critical points. Nonetheless, we will take z* to be the critical point associated with the principal value for ln(τ¯) and consider it to be the principal critical point. Notice that the

Dynamics of the principal critical point

Fig. 1 shows the behavior of principal critical point for q=5 as the complex parameter α is varied. The center of the image corresponds to α=0.63+1.67i and the width of the image is 4.5. For each α corresponding to a pixel position, if iteration of the critical point becomes large (1e5), then we consider it to have escaped and those points are shown in grayscales, with black being rapid escape and white being slow escape. Color (hue) in the portion that remains bounded specifies periodicity of

Julia sets

Filled-in Julia sets are created by taking a fixed function and considering the escape/convergence time for initial points that correspond to screen position. Here, we fix α and q in order to get a specific function to iterate. In Fig. 5 we show the escape and/or convergence time for the filled in Julia set associated with α=−1.03038+1.68012i and q=5 that comes from a distorted green bud seen on the left of Fig. 1. Here the window is centered at −9.4+9i with a width of 14. The green region

Conclusions

Generalized Binet functions have a principal critical point that allows for critical point escape time images to be created that, like the classic Mandelbrot set, are a guide to visually dramatic Julia sets. We have seen that for low values of q, images of the critical point dynamics show distortion that carries over to the corresponding Julia sets. For larger values of q, the critical point dynamics show much more symmetry, but remain more complicated than the classic Mandelbrot set. In all

Acknowledgments

This work was accomplished while Professor Chen Ning was a visiting scholar at Lafayette College. The support Natural Science Foundation of Liaoning province of China (20032005) and the Foundation of Science and Technology Bureau of Shenyang (200143-01) and the hospitality of Lafayette College are greatly appreciated.

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