Markus-Lyapunov Fractals

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Markus-Lyapunov Fractal
Markus-Lyapunov fractals are representations of the regions of chaos and stability over the space of two population growth rates.

Contents

Basic Description

The Markus-Lyapunov fractal is much more than a pretty picture – it is a chart. The curving bodies and sweeping arms of the image are a color-coded plot that shows us how a population changes as its rate of growth moves between two values. All the rich variations of color in the fractal come from the different levels of stability and chaos possible in such change.

The logistic map is one of the most concise mathematical representations of population growth. Depending on the rate of fecundity used in the map, it will generate either a neutral system, a stable oscillating system, or a chaotic system. To help determine which of these outcomes will occur, the mathematician Aleksandr Lyapunov developed a method for comparing changes in growth rate in order to calculate a value called the Lyapunov exponent. This is a useful indicator because, for the logistic map,

  • If it is zero, the population change is neutral; at some point in time, the population size reaches a fixed point and remains there.
  • If it is less than zero, the population will become stable. The more negative the number, the faster and more thoroughly the population will stabilize.
  • If it is positive, the population will become chaotic.
Another example of a Markus-Lyapunov fractal, this one with chaos in black and stability in gold.
Another example of a Markus-Lyapunov fractal, this one with chaos in black and stability in gold.

What does all this have to do with the fantastical shapes of the Markus-Lyapunov fractal? The scientist Mario Markus wanted a way to visualize the potential represented by the Lyapunov exponent as a population moved between two different rates of growth. So he created a graphical space with one rate of growth measured along the x-axis and the other along the y. Thus for any point (x,y) there is one specific Lyapunov exponent that predicts how a population with those rates of change will behave. Markus then created a color scheme to represent different Lyapunov exponents – one color represents positive numbers, and another represents negative numbers and zero. This second color he placed on a gradient from light to dark, so that lower negative numbers are lighter and those closer to zero are darker. The bands of black that appear in many fractals therefore show where the Lyapunov exponent is exactly zero, and bands of white indicate superstable points. By this code, Markus could color every point on his graph space based on its Lyapunov exponent.

Consider the main image on this page. The blue "background" shows all the points where the combination of the rates of change on the x and y axes will result in chaotic population growth. The "floating" yellow shapes show where the population will move toward stability. The lighter the yellow, the more stable the population. Based on this color assignment, if a logistic system with rates of change x = 2 and y = 1.6 has a Lyapunov exponent of -0.3, there will be a dark yellow pixel at the graph location (2, 1.6), showing that the system moves slowly toward stability. If, instead, that logistic system had a Lyapunov exponent of 1, that same pixel would be blue, showing chaos.

A More Mathematical Explanation

Note: understanding of this explanation requires: *understanding Logistic Bifurcation

The Lyapunov Exponent

The Lyapunov exponent is a measure of the rate of divergence of two infin [...]

The Lyapunov Exponent

The Lyapunov exponent is a measure of the rate of divergence of two infinitesimally close points in a dynamic system. For a single-variable system such as the logistic map, we can consider two points at an arbitrarily small distance, dx0 from each other. After n iterations of the system, they will be at distance dxn from each other. The Lyapunov exponent λ represents this change with the approximation:

Eq. 1        {\operatorname{d}x_n\over\operatorname{d}x_0}\approx 2^{\lambda n}

Or, isolating the Lyapunov exponent λ:

\frac{1}{n}\log_2{\operatorname{d}x_n\over\operatorname{d}x_0}\approx \lambda

Generalizing this for all n, we use summation notation to consider every step of iteration:

Eq. 2        \frac{1}{n}\sum_{1}^n \log_2 {\operatorname{d}x_{(n+1)}\over\operatorname{d}x_n} \approx \lambda

In order for this to be accurate, however, it must measure the system's divergence over an infinite period of time, so we define λ as the limit of Eq. 2 as n approaches infinity:

Eq. 3        \lambda=\lim_{N \to \infty}\frac{1}{N}\sum_{n=1}^N \log_2 {\operatorname{d}x_{(n+1)}\over\operatorname{d}x_n}

This is the discrete form of the Lyapunov exponent.

To consider the implications of λ, let us return to Eq. 1:

{\operatorname{d}x_n\over\operatorname{d}x_0}\approx 2^{\lambda n}

We can see that, for λ < 0, the difference between dx0 to dxn will disappear as n grows. (Indeed, the lower the value of λ, the more quickly this change will disappear.) Similarly, for λ = 0, there will be no difference at all. For λ > 0, however, the difference between the initial distance between the points and the final distance between the points will expand exponentially as n grows. In other words, a positive Lyapunov exponent indicates a system in which an infinitesimal change in initial conditions can result in massively different final conditions. A negative Lyapunov exponent, on the other hand, indicates a system in which the effects of such an initial change will fade over time.

Recall that the phenomenon captured by a positive Lyapunov exponent – wide variation resulting from infinitely small initial changes – is one of the conditions for a system to be chaotic. Thus a positive Lyapunov, in the presence of other conditions of chaos, implies a chaotic system.

In the Logistic Map

Recall that the logistic map is:

Eq. 4        x_{(n+1)}=\mathbf{r}x_n(1-x_n)

From which we find

{\operatorname{d}x_{(n+1)}\over\operatorname{d}x_n}=\mathbf{r}-2\mathbf{r}x_n

Inserting this into the Lyapunov exponent, Eq. 3, we have the Lyapunov exponent for the logistic map:

Eq. 5        \lambda=\lim_{N \to \infty}\frac{1}{N}\sum_{n=1}^N \log_2 \mathbf{r}-2\mathbf{r}x_n

As noted above, negative values of λ here indicate stability in the logistic map. Also, in the case of the logistic map, any system with a Lyapunov exponent greater than zero is a chaotic system. We can see this when we compare the logistic bifurcation diagram with a graph of the Lyapunov exponents for the logistic maps of changing r values:

Here a graph of the Lyapunov exponents for logistic maps with 0 < r < 4 is overlaid in red on the logistic bifurcation diagram.  The points of the red graph were calculated directly in Matlab using Eq. 5.
Here a graph of the Lyapunov exponents for logistic maps with 0 < r < 4 is overlaid in red on the logistic bifurcation diagram. The points of the red graph were calculated directly in Matlab using Eq. 5.

Forcing the Rates of Change

Mathematically, the important part of Markus's contribution to understanding this type of system was not his method for generating fractals, but his use of periodic rate-of-change forcing. The value of r has a great impact on the output of the logistic map, but this value can have still greater impact if we do not choose to keep it constant.

A Markus-Lyapunov fractal with rate-of-change pattern ab
A Markus-Lyapunov fractal with rate-of-change pattern ab

In terms of the logistic map, this means we choose a set of rates of change, r1, r2, r3,..., rp, where p is the period over which the rates of change loop. When we force the rates of change to follow such a loop, we have a new, modular logistic map (Eq. 4):

Eq. 6        x_{(n+1)}=\mathbf{r}_{n \text{mod} p}x_n(1-x_n)

It is in these forced alterations in rates of change that the fascinating shapes of the Markus-Lyapunov fractal come out. Each of the fractals is formed from some pattern of two rates of change, a and b. So a pattern aba would mean each point on the fractal is colored based on the Lyapunov exponent of the logistic map Eq. 5, where r1 = a, r2 = b, and r3 = a. That is, the r values would cycle a,b,a,a,b,a,a,b,a....

Because the axes used to map these fractals are measurements of changes in a and b, the pattern a would simply yield a set of vertical bars, just as the pattern b would yield horizontal bars. However, once the patterns start to become mixed, more interesting results come out. The image to the right shows an ab pattern. Note that it is much simpler, in the quantity of spires and crossing arms, than other images shown on this page; the main image, for instance, is a bbbbbbaaaaaa pattern.


Why It's Interesting


An enlargement of a section of "Zircon Zity," showing self-similarity.
An enlargement of a section of "Zircon Zity," showing self-similarity.

Fractal Properties

The movements from light to dark and the dramatic curves of the boundaries between stability and chaos here create an astonishing 3D effect. But the image is striking not only for its beauty but also for its self-similarity. Self-similarity is that trait that makes fractals what they are – zooming in on the image reveals smaller and smaller parts that resemble the whole. Consider the image to the right, enlarged from a section of the main image above. Here we see several shapes that repeat in smaller and smaller iterations. Perhaps ironically, this type of pattern is a common property of chaos.

For more images of the fractal properties of chaotic systems, see the Henon Attractor, the Harter-Heighway Dragon Curve, and Julia Sets.


One artist superimposed and edited several real Markus-Lyapunov fractals to create this piece of art.
One artist superimposed and edited several real Markus-Lyapunov fractals to create this piece of art.

Artistic Extensions

After Markus saw the incredible beauty and intriguing three-dimensionality of the images generated by his plotting system, he immediately sent the images to a gallery in the hopes that it would display his images in an exhibition.[1] It's easy to see why he did so, and in fact, pictures based on these fractals have become a large part of what is called "fractalist" art. As with all domains of fractalist art, there is a great deal of debate in the art community over whether these images are truly "art" given their intrinsic reliance on a purely scientific, algorithmically-generated chart. One could say that such a process is devoid of creativity, but it is equally valid to say that the identification and presentation of the beauty in the science is an art in itself – a concept that is critical in modern art. Either way, there has been an undeniable artistic fascination with Markus-Lyapunov fractals; if the image seems familiar, you have likely seen it on posters, t-shirts, or any other canvas for graphic design.


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References

  1. Dewdney, A. K. (1991). Leaping into Lyapunov Space. Scientific American, (130-132).





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