Views of Fibonacci dynamics

Abstract

The Binet formula gives a natural way for Fibonacci numbers to be viewed as a function of a complex variable. We experimentally study the complex dynamics of the Fibonacci numbers viewed in that manner. Attracting and repelling fixed points are related to the filled Julia set and to regions of escape time images with fascinating behavior.

Introduction

The Fibonacci numbers are traditionally described as a sequence F_n defined by F₀=0,F₁=1, and F_n=F_n−1+F_n−2. The sequence begins as 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,… .

The Fibonacci sequence has many remarkable properties, ranging from routine to startling [1], [2], [3], [4]. Moreover, the numbers arise in nature, for example, as the number of spirals of pinecone petals. They may also be used to construct mathematical quasicrystals [5].

One of the beautiful formulas of Fibonacci numbers is the Binet formula. Binet described a version of the formula in 1843 [6], [7]. Its beauty arises from the fact that the formula gives a closed form solution to a recursive definition, and from the symmetry of the formula itself. The Binet formula may be derived from the theory of difference equations, it can be derived by diagonalizing a suitable matrix, or it can be proven by induction [1], [2], [3]. The Fibonacci recursion has characteristic equation x²−x−1=0 which has roots $\text{τ=1+}\text{5}\text{/2≈1.618}$ and $\text{τ}\text{̄}\text{=1−}\text{5}\text{/2≈−0.618,}$ where τ is the golden ratio and $\text{τ}\text{̄}$ is the conjugate of τ. Choosing constants to satisfy the initial conditions F₀=0 and F₁=1 gives the Binet formula: $\text{F}{}_{\mathrm{n}}\text{=τ}{}^{\mathrm{n}}\text{−}\text{τ}\text{̄}{}^{\mathrm{n}}\text{/}\text{5}\text{.}$ To obtain the Fibonacci numbers as a function of a complex variable, instead of viewing the index n in the Binet formula as an integer, we view it as a complex variable z. Thus, we define the following complex Fibonacci function:

$$\text{F(z)=}\text{τ}{}^{\mathrm{z}}\text{−}\text{τ}\text{̄}{}^{\mathrm{z}}\text{5}\text{.}$$

The number $\text{τ}\text{̄}$ is negative and $\text{τ}\text{̄}$ appears as the base of an exponential in the Binet formula. Thus, complex numbers will result for fractional real arguments. Nonetheless, the Binet form gives a natural generalization of the Fibonacci sequence. It satisfies the initial conditions F(0)=0 and F(1)=1. It also satisfies the recursion F(z)=F(z−1)+F(z−2) and it is defined for all complex values of z.

Thus, we can ask questions about the complex dynamics of this function. What are its fixed points? Are they attracting or repelling? What happens upon iteration of the function? In this note we take a visual look at those questions and see that the Fibonacci numbers have interesting and beautiful complex dynamics.