SIMPLE CHAOS THEORY
Can chaos be explained in a very fundamental way, without resorting to Hamiltonians and phase space, to give an intuitive feel for what
is going on? This is attempted here.
Chaos theory is to be found in many places from the giant red spot on Jupiter to dripping taps, and in the biological realm in heart fibrillation and brain seizures. Feigenbaum discovered a way of describing it, although he was not the first to discover chaos, it being known to Einstein, and even before him in the 19th Century from the study of dynamical systems where phase-space orbitals could cease to be well defined. It was largely ignored until the meteorologist Lorentz found that his simple model of the atmosphere did not give repeatable results. The advent of the PC with sufficient power to implement chaotic systems finally opened up the subject to wide research and application, although we might recall that Feigenbaum used a simple calculator to make his initial discovery! The actual existence of chaos as a fundamental fact rather than a mere appearance arising from inadequate precision in the calculations interested the engineer writing this. In other words he was sceptical: was it just 'hype'? What is actually happening is not easy to grasp from the advanced maths used. Below we show the classic figure for the equation y=rx(1-x) when handled recursively i.e. the calculated value of y is re-inserted as x in the equation, and so on. The value of r is increased from 1 to 4 along the x-axis.
Ignoring the asymptotes, the function appears as a single line on the left where the recursions converge on a single value. As r is
increased a bifurcation is reached at r = 3, the two resulting lines continuing toward the right until two more bifurcations occur
(a so-called period-doubling) at r = 3.449, and so on. The dense blue regions contain regions of genuine chaos mixed with reversions
to non-chaos. In brief, what happens is that the interval between period doublings decreases as r increases, tending to zero before
r reaches 4, at which point there are infinitely many bifurcations, and we have chaos. Reversion to non-chaos occurs when the equation
cycles finitely for reasons we cannot explain briefly. An exploration of this together with a justification that chaos does exist
fundamentally is explained in the article IS CHAOS GENUINE? which may be downloaded. It is a ZIP file
containing three WORD files, one containing diagrams.
The tetrahedral complex is introduced in the archive article TETRAHEDRAL COMPLEX and it was found that chaos occurs within projective geometry itself when polarity is traversed recursively in a tetrahedral complex. The picture below shows a diagram for this chaotic polarity which is its equivalent of the famous Mandelbrot set.
The colour codes for the number of iterations before the cubic function goes to infinity are shown on the right. This is only a portion of the whole set which extends to infinity. On the upper left there is a 'fractal bridge' between two 'globs', which looks the same at all magnifications, reminiscent of God reaching his finger towards Adam. The true fractal nature of the process is illustrated by the following picture taken from within the vertical strip:
The equation relating polar lines in the complex which when iterated leads to the above pictures is
where lambda is the cross-ratio in which a line cuts the tetrahedron and k is the fundamental cross-ratio defining the complex.