Within this piece of writing, you will be able to see some amazing and even eerie relationships between mathematical concepts that seem distant.
But above all, what is a logistics function?
Logistics functions are functions in the form of
Its mathematical significance is that it is one of the simplest yet most widely applicable functions that has a negative feedback loop, that is to say, its previous value restricts the next value. This function often can be used to model populations, since population sizes are restricted by limiting factors.
In fact, one could approach the recursive nature of the function in a graphical way (this is what the video doesn’t show you). After drawing the function for x_t, draw the line y=x.
But this function becomes magical when one continually increases the value of r. At first, the function settles at a single value. However, as r increases, the function settles down at two values, instead of one, meaning that it oscillates between the two values. Then, as r continues to increase, the values become four, eight, sixteen, and finally when r approaches 3.8, one suddenly sees chaos. What is even more baffling, is that if one continues to increase the value of r from 3.8, one will sporadically have a function that oscillates between 32 values. Soon, the function again becomes chaos. The function once again retains stability with 64 values, and again becomes chaos. This pattern is perpetuated throughout. This phenomenon is known as bifurcation (fig. 1), where the function oscillates between 2^n values.
fig. 1. Magical Bifurcation. Note that on the x-axis is the variable r
Frankly, I am unsure why the value is around 3.8, and the number 3.8 is even a value that doesn’t bear many connotations in math. I tried to find the exact value of the point where chaos happens but failed.
What is even more amazing is how often this formula is found in fractals. The Mandelbrot set, one of the most famous set of complex numbers that form a fractal, also bears a huge relationship with this formula. The Mandelbrot set is defined by the equation
fig. 2. The Mandelbrot set in the imaginary plane
and has the following shape in 2D (fig. 2). However, if one was to see the diagram in 3D, the graph of the Mandelbrot set in the z dimension forms the exact same shape as in fig. 1. (fig. 3)
fig. 3. Bifurcation and Mandelbrot Set together
In fact, we can rigorously prove this quite simply. If we reflect the logistics function by the x axis, and translate the graph by 1/2, we get an exactly identical form as the generating function of the Mandelbrot set, only that the C is r/4.
But the craziest part of the equation is yet to come: the equation is even found in nature—also human beings ourselves. When one puts liquid mercury into a rectangular cylinder, heats it gently and uniformly from underneath, and measures the temperature at a set point, one would get the oscillating form of the logistics function graph. Even more, put r as the temperature of the base of the cylinder, and one would get the entire graph as the temperature of the base of the cylinder.
Also, the equation also exists in us humans—as light intensity increases, the time taken until we close our eyes is also identical to the logistics function. I posit that this is because the diffusion of neurotransmitters between neuron junctions can be modelled by the logistics function. Another aspect is in the heart; when one suffers a heart attack, the heartbeat rate is identical to the values of the logistics function, when time after the seizure is put as r. Even more, a group of scientists have used this fact to predict how the heartrate will differ, and used electric signals to correct for the errors of the heart. Whether this is just a baffling coincidence or an intricate phenomenon is still unclear. But the latter seems more likely and clear.
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