The Mandelbrot Set and the Logistic Map
The central equation of the Mandelbrot Set
and that of the Logistic Map
have obvious similarities, in that they are both iterative relationships based on quadratics. The detail looks different, in that the Mandelbrot Set equation has no linear term, and the Logistic Map has no constant. But a suitable substitution can solve those issues, and putting
results in a rescaled Logistic Map with no linear term but with a constant.
So let us consider the Logistic Map formed by the equation
and we will plot this for -2<x<0.5 underneath the familar image of the Mandelbrot Set.
Should this image surprise us? Perhaps not. Consider the following points.
It looks like a rescaled, stretched and inverted version of our original Logistic Map. So it is. We have given the relationship between the x of the orginal map and the x of this map, but not that between the old λ and the new c.
It shows what happens to points on the real line as the Mandelbrot formula is repeatedly applied. We know that for -2<x<0.25 they are in the Set, so do not diverge to infinity. The Logistic Map gives us more detail of how they behave.
For -0.75<x<0.25, that is the main body of the set, the iterations converge to a single value. That the point at zero stays at zero is obvious. Another example would be the point at 5/36. This converges to 1/6, where (1/6)2+5/36=1/6. Similarly the point at -5/16 converges to -1/4, so when x=-0.3125, y=-0.25, on our rescaled Logistic Map graph. (Proving that -0.25 is a fixed point when c=-5/16 is easy. Showing that it is stable, attracting nearby orbits, (another way of saying that nearby values will converge to it) is slightly more difficult.)
For -1.25<x<-0.75, the head of the set, the iterations oscillate between two values. Most obviously the point at -1 cycles between -1 and (-1)2-1=0, from which it returns to (0)2-1=-1.
The next section shows the iterations converging to cycling between four values. It corresponds to the next largest bulb on top of the head and extends to about -1.36. An example with a period of four can be found at c=-1.3107 (approximately), which converges to the cycle -1.3107, 0.4072345, -1.14486, 0. Here c is a member of the final cycle, but it does not have to be so.
The length of the cycle, having doubled from one to two to four, now doubles increasingly rapidly, and soon becomes infinite by around -1.4.
A region of order appears in the Logistic Map, in which it cycles between just three values. This period three region, after our rescaling, appears at about -1.75. This is precisely where the biggest of the satellite Mandlebrot Sets is to be found as one moves along the thread joining the features found on the real line back to the main set which stops at about -1.4. There is a smaller period 5 window in the Logistic Map at about -1.625, and this too corresponds to a larger-than-usual satellite set on the real line, although it is not marked in the above figure. The other moderately-significant satellite corresponds to a period 6 window in the Logistic Map. Note that these period lengths of three, five and six are not in the sequence formed by the initial period doubling giving one, two, four, eight etc.
Most of the Mandelbrot Set codes on this site default to plotting in a window 600 pixels wide. So for convenience when generating the figure on this page the Logistic Map code was adjusted to use the same scale: logistic_mandel.py. This code looks very basic and unoptimised beside some of the Mandelbrot codes, but it is arguable that it is good enough, so why spend time making it better?
Finally note that although the Mandelbrot Set along the real axis is simple, in that -2<=x<=0.25 is in, and all other points out, so it is a single line of rational length, the Mandelbrot Set along the imaginary axis is very different. It starts simply enough, with all points with |z|<0.63... being in, but then it becomes very complicated. The last point on the imaginary axis to be in is at ±i, where, for +i, the formula iterates to -1+i, then -i, then back to -1+i, ending in this cycle of period two.
Between just over 0.63i and i there are other points which are in the set, although most are not. It is not clear to me whether these other points are all isolated points, or whether there is another region of non-zero extent on the imaginary axis which lies within the set.
Note too that the point at z=-2 is the point with the greatest modulus of real component (and also the point furthest from the origin). The point at z=i may be the furthest point from the origin on the imaginary axis to be in the set, but it is not the point with the greatest imaginary component. There is a tendril which stretches up to around -0.207108+1.122757i.
Similarly the point with the greatest positive real component is not that on the real line at 0.25, but rather at about 0.471+0.354i.