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Several years ago, the Great Mandelbrot Fever began after the Scientific American revealed several of the beautiful Mandelbrot pictures obtained by H.O. Peitgen and P.H. Richter, later published in "The Beauty of Fractals" (Springer-Verlag, 1986). Could the same story have happened again? Very appealing pictures appeared in the September 1991 issue of the Scientific American, again with a (seemingly) simple numerical recipe to get them. Such a picture is featured just above, and a few more at the end of this paper. How about hunting for such pictures by yourselves?

Unfortunately, there is a serious problem: by 1992, with a 25 Mhz 68030, interesting images took ages to be computed, up to ten hours or even more. The 1997 Pentium and PowerPC are far faster but one now asks for increased resolutions and interesting images, not already explored, are more difficult to compute, so these fractals still require hours of CPU.

In a nutshell, the pictures are nothing but a mapping of Lyapunov exponents relative to a simple extension of the logistic formula (this extension is due to Mario Markus, hence the name Markus-Lyapunov). Want more information? Just read on! And don't let the following few notions of contemporary mathematics deter you!

The logistic formula

The logistic formula (also known as "Verhultz dynamics") is the simplest-known formula describing a chaotic dynamic system. Maybe you don't even know what a dynamic system is? Generally speaking, it's a highly abstract concept: think of a "system" which is examined at successive times; at time (n), the system is described with a variable X(n) (possibly multidimensional) and one admits that a specific law exists that allows to derive X(n) from the preceding X(n-1) (and the latter only). The object of the theory is to discuss the behavior of the successive X's when n grows to infinity. Do they converge towards a unique, stable value? If not, do they behave "regularly"? Or do they have a seemingly erratic, unintelligible, "chaotic" behavior?

The logistic formula simply is a rudimentary attempt to modelize the evolution of an animal population. Here the variable X(n) is the number of animals in an isolated region, at year (n). If the growth rate was constant, the population at year (n+1) would be proportional to the population at year (n), namely

X(n+1) = R * X(n) .

where R would be constant. Unfortunately, such an assumption would lead to an unrealistic exponential behavior. In fact, the growth rate must decrease when the population increases, because there is less and less food available. The simplest way to express this idea consists in assuming that R at year (n) can be put in the form R0-a*X(n) or, equivalently,

R = r * [1-X(n)/Xmax]

where r and Xmax are two constants, unspecified so far; Xmax clearly is an upper bound for the population, beyond which the calculation becomes meaningless (the next X would be negative, or zero); r is called "fecundity factor" (the larger it is, the larger the increase from one year to the next one). Putting x(n)=X(n)/Xmax, the evolution law finally becomes

x(n+1) = r * x(n) * [1-x(n)]

This is the logistic formula. Interesting x values lie between 0 and 1.

[ image ] The analysis of this "dynamic system" -i.e. the observation of x(n) when n goes to infinity- for various values of r, can be done with the figure on the right. For every point on the r axis, the logistic formula was applied 200 times, starting from x(0)=0.5 (in fact, this value does not matter), and then 300 times again, but now with plotting the corresponding (r,x) pixel in the figure each time. One notices that there is a single pixel for every r lower than 3: this means that the population then converges to a stable value.

Every r between 3 and 3.45 corresponds to 2 pixels: this means that the population oscillates between a low value (with plenty of food available and a rapid growth), and a high value (no food enough, hence increased mortality): the "fecundity" is too high for a unique stable equilibrium to be possible. Things become more complex when the fecundity is further increased. First, the population oscillates between 4 values, then between 8, 16... values, and for r > 3.57 it seems that there is an infinite number of possible points (forming continuous areas in the figure), between which the population seems to oscillate randomly: this is the chaotic behavior.

[ image ] However, the transition to chaos is not definitive: you can see islands with no dark pixels, corresponding to a quieter evolution of the population, which again alternates between a small number of values. You can see the details of such a stability island on the left; the whole figure is an enlargement of the marked area in the preceding image. Notice that the two figures are strikingly similar: this "self-similarity" is a common feature with fractal pictures.

Lyapunov exponents

Lyapunov was a Russian mathematician who lived around the turn of the century, long before computers were invented. We now explain his part in our story.

We have seen that, for low fecundity (r < 3), the population modelized with the logistic formula converges towards a single value. This limit does not depend on the initial value. If the population is suddenly modified at a given time, it will return to this limit after a few turns. A similar behavior can be observed for multi-valued oscillations: the 2, 4, 8... values forming the limiting cycle do not depend on the initial population, and this cycle is always recovered after a sudden perturbation. However, this return to the periodic cycle is all the longer as one is nearer to the chaos threshold. The limiting cycle is then said to be less and less "stable". This stability can be characterized with a number, the Lyapunov exponent. Stable evolutions correspond to large negative exponents; the stability decreases as the exponent goes closer to 0; finally, positive exponents correspond to chaotic evolutions.

The theory of Lyapunov exponents is not easy. The Scientific American paper gives only a practical recipe for computing them in the case of the logistic formula. Take it as a magic formula:

  (1) Initialisation  x = x0                                                
                      For i=1 to INIT:  x = rx(1-x)

  (2) Derivation      total = 0        
      of the          For i=1 to NLYAP: x = rx(1-x) 
      exponent                          total = total + Log|r-2rx|/Log 2
                      exponent = total / NLYAP 

x0 is arbitrary (between 0 and 1); INIT and NLYAP are arbitrary integers, which must be as high as possible to get precise evaluations.

The Markus idea

Mario Markus (from the Max Plank Institute for Nutrition) has imagined dynamic systems just slightly more complex than the logistic formula, where the fecundity is forced to alternate between two values r1 and r2, following a periodic pattern. The period is described by a root string made with "1" and "2"; for instance, a root "112" means that r takes successive values r1,r1,r2, r1,r1,r2, r1,r1,r2....and so on. These systems also display stable limit cycles or chaotic evolutions, depending on r1 and r2. The stability or the chaos can be analysed by computing the Lyapunov exponent, through the above recipe, where the only change is that r is now forced to follow the prescribed periodic pattern.

[ image ] The Markus pictures are simply a color mapping of the Lyapunov exponent versus r1 and r2, along horizontal and vertical axes respectively, for a given root string, here "1122".

Only the stability domain is plotted; here, chaos (i.e. positive exponents) is rendered in dark blue. When the exponent goes from 0 to minus infinity, shades range from light to dark. At zero, (chaos threshold), the color suddenly jumps from dark blue to a light shade. Obviously, this color mapping is arbitrary and can be changed as you want.

One thus gets strange shapes floating in a fantastic universe, with a striking 3-D aspect. Of course, the pictures here are too small. You must see them in a full screen to do them justice.

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The rest of the game is the same as in classical Mandelbrot explorers: you start from a general picture; you look for promising details and you enlarge them. For instance, the larger rectangle in the above picture leads to the picture on the right; the smaller one leads to the picture in the title of this article. Click here to see an example of research for a complex root.

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This last picture was reused by the author in the composition "Image 11". It was computed with the root 11212, INIT=400 et ITER=800. The reward for a lengthy calculation...

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e-mail Charles.Rosemarie.Vassallo@wanadoo.fr