Can complex numbers, chaos theory, fractals, and power laws exist independently?

[Justice Hunter]

What is the connection between complex numbers, chaos theory, fractals, and power laws?

By connection i mean, does one require the other in order to exist?

For example, from my readings, complex numbers gave rise to the chaotic system, that proceeded to create the Mandelbrot set.

So the question is are each a requisite to the next? Or can any of these develop without the prior? Do these arise in our current understanding of modern physics?

 

[HallsofIvy]

No, we can have chaotic systems and the Mandelbrot set without complex numbers. For example, "start with a number, x, between 0 and 1. At each step double x then drop the integer part and keep only the fraction part". For example, if we start with, say, x= 2/3, then 2x= 4/3 so, dropping the integer part, we have 1/3. Doubling again, 2/3 again and then it repeats. That gives sequence, 1/3, 2/3, 1/3, 2/3, ... with "period 2". And, in fact, we can get sequences of any period that way so this is a chaotic system in the real numbers..

The only reason complex numbers come into it is that we get "nicer" pictures if we work in two dimensions and the complex numbers are a two dimensional set.

 

[SteamKing]

There's nothing chaotic or complex about the power laws, except how some people apply them occasionally.

 

[WWGD]

The iteration of the Logistic equation gives rise to Chaos , without need of Complex Numbers. A fractal, meaning a space with non-integer Hausdorff dimension (Edit: seems some describe it as a space whose Hausdorff dimension is larger than the topological dimension) can happen without use of Complexes too: the Cantor set has non-integer Hausdorff dimension log2/log3. I understand a chaotic system to be a Dynamical system whose attractor set (a version of a limiting space/set) is a fractal.