Tag Archives: Complex Quadratic Functions

Fractals & Chaos Recap for 1/10

We finished proving some of the observations made yesterday, and also discussed an argument that all Julia sets are either completely connected (any two points can be connected by a path along the Julia Set) or completely disconnected (every point is disconnected from every other point). This difference gives us a convenient way to categorize Julia Sets, allowing us to create a catalog view of them, much like we did for the Feigenbaum Plot in the real numbers.

What we need, then, is a convenient way of identifying whether or not a Julia Set is connected without having to actually draw it. Fortunately, iterations of points contained within the Julia Set give us a way to do that: if we can find seeds that do not diverge to infinity, then the Julia Set is connected. More specifically, we have argued that the origin, z = 0, is a convenient starting seed to iterate. If the orbit of z = 0 eventually tends to infinity, we know the Julia Set is disconnected. But how do we know the path of an orbit will actually tend to infinity and never return?

Fortunately, there’s a radius of no return. We further proved in class that if the path of an orbit of the origin exceeds r = 2the orbit will never come back. This means that any value of |c| > 2 results in a Julia Set that is automatically disconnected (since if z_0 = 0, then z_1 = 0^2 + c = c, and we’re already past 2), and for any value of |c| < 2, we just need to iterate until the orbit exceeds 2.

If we imagine the complex plane as a computer image, where every pixel corresponds to a single value of c, then we can colorize those pixels based on whether or not their orbit has “escaped”. As we increase the number of iterations, more and more pixels will become colorized (we can use the same colors for pixels that escape at the same number of iterations). Eventually, we will see a shape form. That shape is the Mandelbrot Set.

Download and investigate the Mandelbrot Set using the program Fractal Zoomer. Make sure you download Fractal Zoomer.exe (unfortunately, this software only functions on Windows-based computers). There are three modes in the basic view screen: Zoom Mode, Julia Mode, and Orbit Mode

  • In Zoom Mode:
    • Left Click = Zoom in
    • Right click = Zoom out
    • Ctrl+F3 = Set new center
  • Press J to activate Julia Mode
    • Left Click anywhere in the Mandelbrot Set to create the Julia Set at that value of C
    • Ctrl+F3 will allow you to spawn a Julia Set of a specific value of C (use this to investigate Julia Sets from the Complex Paint Worksheet)
  • Press O to activate Orbit Mode
    • Left click anywhere in the Mandelbrot Set and it will superimpose the orbit of seeds within the Julia Set that uses that value of C
    • Ctrl+F3 will allow you to specify the orbit of a particular value of C (or if you are in a Julia Set, of a particular seed).

Fractals & Chaos Recap for 1/9

We spent some time reviewing the results from Complex Paint Worksheet 2. We have observed a few things, the most obvious of which are that all Julia Sets have 180-degree rotational symmetry and the only Julia Sets that have x- and y-axis symmetry have a parameter that is a real number. It turns out that both of these observations are provable facts about Julia Sets, which we proceeded to start, and will finish tomorrow.

Fractals & Chaos Recap for 1/7

We discussed the results from iterating yesterday’s seeds in the function z², finding that some seeds will attract to zero, some will spiral off to infinity, but others seem trapped in a unit circle around the origin, falling into a cycle or landing on (1,0). This unit circle forms a boundary between seeds that diverge and seeds that do not, and that boundary is called a Julia Set, named after French mathematician Gaston Julia, and developed by Julia and Pierre Fatou (Fatou names the complement of the Julia Set; in the case of C = 0, the Fatou set is the entirety of the rest of the complex plane except for the unit circle).

Not all Julia Sets (in fact pretty much none of them) look as simple as this, and Complex Paint is a great tool to help us see them. Watch this video for instructions on the derivation of Julia Sets, plus instructions on how to use Complex Paint to create and understand this new class of mathematical object. Feel free to explore whatever parameter values you would like, but in particular we will be using the ones found on Complex Paint Worksheet 2. We started this today and will finish it tomorrow. For each, note the type of Julia Set you get (area, string, or dust), any symmetry you observe, and the destination of orbits inside any area Julia Sets.

Fractals & Chaos Recap for 1/6

We started today with a brief recap of the work we’ve done with complex linear functions using the back of the Complex Paint worksheet we’ve been working off of.

From there, we started work with complex quadratic functions. We will be replicating the process we did with real numbers: analyzing a family of quadratics where we only adjust a single parameter value and investigate the behavior of iterations for specific values of the parameter. Eventually, we will find a way to categorize and catalog these behaviors.

The family of functions we will be analyzing is  + C, and the first parameter we will look at is C = 0. We found that squaring a complex number in polar form resulted simply in squaring the value of R and doubling the value of θ, so an initial seed like [2,10°] becomes [4,20°], then [16,40°], then [256,80°] and so on, for larger and larger values of R, suggesting that the seed [2,10°] diverges to infinity under iterations of z².

If you weren’t in class, pick two of the seeds below to iterate. Iterate until you’re convinced what the long-term destination might be (diverging? converging? limit cycle?). Note also that θ should never exceed 360°. If the pattern for [2,10°] from above were continued, we would get 160°, then 320°, then 540°. But 540° is larger than 360°, and is equivalent to 540-360=280°. And that’s the angle we would record.

  • [1,45°]
  • [1/3,60°]
  • [1,30°]
  • [5,180°]
  • [1,7°]
  • [1,120°]
  • [1/2,36°]
  • [2,90°]
  • [1,10°]