Several years ago, the Great Mandelbrot Fever began after the
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 (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.
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
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 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 theory of Lyapunov exponents is not easy.
The
(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
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. 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.
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.
This last picture was reused by the author in the
composition "Image 11". It was computed
with the root 11212, Charles.Rosemarie.Vassallo@wanadoo.fr |