The beauty of Mathematics and Science.....

 

Ever wondered what is this pattern? Some of have seen this pattern get zoom into and then repeating itself from the small outgrowths present and it continues as if there is no end to this..

This is called as the MANDELBROT Set. This set is characterised by a set of complex numbers that come in the equation of , which does not diverge if z=0, i.e. for the function f(0) and so on.

For each value of c the set becomes more stable. and thus sometimes cannot remain stable.

There is something common between the reproduction of the population, a leaking tap water, and the human heart beats and neuronal transmission.

The amount of times a population can grow is just a multiplication factor, for instance if u know that the population of rabbits double every year then u can easily predict the population next year. If u plot the graph of population of this year and next year, u get an inverted parabola (something similar to shown in graph below).

Now if u know the growth factor u can always know that the population can be in equilibrium, but if the value of this growth factor reduces, it causes extinction of the population. So now if we plot the graph of the population at equilibrium vs the multiplicative factor, with the factor starting from lowest value, we see something interesting.

The graph slowly rises as the inverted parabolic curve, but when the population reaches to the factor of 3, the graph splits in two. It constantly oscillates between being high reproductive and sometimes low reproductive, but never settles at a constant value. As time progresses this becomes more unstable and each fork divides into 2 more creating more disturbances and the graph now looks like this.

This is the ultimate chaos, where the graph splits, stabilises, then again splits, then again stabilises. This stabilization is seen as while lines in between the graph on the right hand side. This is a period doubling cycle which means one cycle take twice the time required than normal. So it goes from 2, 4, 8, 16, 32, 64 and so on till it becomes Chaotic. This chaos is reached when the multiplicative factor reaches 3.57 creating random population numbers and each year there is no particular number. Every year the population is different. 

So if u look at it, it seems like a fractal set that goes on as the scale increases with repetition. 

Also this fractal is a part of the mandelbrot set.

 

From this image, we can clearly see the mandelbrot set and the fractal as a single entitiy, which goes on forever. never ending.

Same thing is observed when there is a leaky tap water. the drops that come out of it follow the same pattern. If u want to try, then let a small drop of water leak through the tap and see the motion, it is stable. You see a the drop coming out at regular time intervals. Then when u open it more, the drops come at a speed but the pattern remains same, it may be 2 drops together or 3 drops or even 5 drops together, but this doesn't change; Until You open it at a certain point where that pattern becomes Chaos and the water comes out as a column and not as droplets.

Another example is seen when u flicker a tubelight in front of the eye of a person. The amount of blinking of the eye depends on the blinking of the tubelight, and the neurons adjust themselves creating a perfect pattern in match with the tube light flickering. but when the flickering goes erratic, you get? Yes a seizure, a state of chaos!

You can see in this image taken from the journal. The pattern is faint, but similar.

So the amount of periodicity or the rate at which the curve divides or the intervals of division is constant no matter what and how. If u zoom in more, the more bifurcations you see, but the number doesn't cross. This constant is called as Feigenbaum Constant and has a value of 4.669.

This constant is derived by dividing the width of the parent prong and then the width of the divided prong.

 As you progress further and further, the bifurcations arrive faster, but the number oscillates between 4.669 and 4.7 but never exceeds.....

In another study, scientists used a drug that sent hearts of rabbits into fibrillation. A fibrillation is a condition where your heart contracts so fast that the blood is unable to move out of the heart and the tissues are not perfused properly.

What they saw is a period of oscillation, like initial it was regular beat, then it became 2 beats, then 4 beats and so on till it reached a state that there was no pattern visible.

But still this same graph was appearing but it was faint, and with this graph they determined when to give the electrical shock to reverse the changes and bring it back to normal. So in medicine, we use CHAOS to bring STABILITY... Isn't that interesting??? 

If u liked this article, please let us know in the comments below.

This was possible after a lot of work and watching videos on youtube with help of channels like Veritasium.

All the images are taken from the research articles and credit goes to their rightful writers and publishers.

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{\displaystyle f_{c}(z)=z^{2}+c}