Chaos and Irrationality

Chaos and Irrationality are two concepts that usually do not occupy the same space. In terms of human behaviour, intuatively people tend to grasp the underlying issues at stake but still refuse to own up to or confront the human drama involved. Whereas in mathematics both subjects have deep wells that tend to become walled silos when researchers fall in.

In the December/January edition of Electronics World two articles on Chaos were brought to our attention from researchers in Turkey: “Simulation and application of Memristor-Based Chua's Chaotic Circuit” and “An Elecrtonic Card for Easy Circuit Realisation of Complex Nonlinear Systems”. Both article's hardly scratch the surface of what is involved. Both also apply standard LabVIEW software tools and oscilloscope measurements. Consequently the Complex Number System is adopted, whether by deliberate intent or otherwise.

Probably like most readers, I was first exposed to the mathemitical concepts involved in Chaos Theory through the James Gleick book “Chaos: Making a New Science”. A universal best seller in 1987. Back then it mearely became the subject of pub-talk for a few interesting technical debates, for the simple reason that Gleick drew on applications of the science of chaotic behaviour from far and wide that he never really homed in on his subject proper. Accordingly one former physics lecturer colleague drew my attention to a single paragraph which Gleick had included that was so out of context, he had obviously just quoted it from an unnamed source. It was the only memorable example I still recall, and no, I do not intend to quote it. However, it was Gleick's omissions from his popularisation task that have always bothered me the most. It has always appeared to me that at the heart of modern Chaos Theory, deeply disguised and hidden away is the mathematical concept of Irrationality.

Without, irrationality in mathematics their would be no chaos theory. Yet pick up any book or article on the subject, including both EW articles, and irrationality does not get a mention. Why not?

The involvement of the mathematical concept of irrationality was brought home to me when exploring the Billiard Table Problem. Chaos and Irrationality is at the heart of this mathematical problem also. Though I have never come across a formal mathematical proof. Thus I would like to set another hare running and hopefully can guide these Turkish researchers to approach their subject from a fresh angle.

Let us begin once again by recognising that not all problems can best be analysed by adopting the Complex Number System. Nor the binary or decimal systems. In this particular field the place to start is with the paper “A Number System with an Irrational Base” by George Bergman from 1957. This is a classic paper within a certain community as it was the first to call attention too the importance of the insights that are to be gleaned by applying a change of base: in Bergman's case the Golden Section as a base. Decades later Bergman's work was formally extended to encompase a Generalised Irrational Number System by other researchers who shall remain nameless. Thus the key for analysing chaotic behaviour is recognising and adopting the correct base. Then all else falls into place.

So, as one of the EW articles stated ”It is of immense importance to build these devices, which will help secure communications for applications that need them”. Nope - I just don't buy it. Choose the correct irrational base and all is decipherable. Of course you would also need the enhanced software tools and instruments for the purpose. But that is a further aspect for NI and Tektronix to sort out.

Barry McKeown

January 2018