Discover the Hidden Sounds of the Logistic Map 🎶

In case you ever wondered what the logistic map sounded like! The logistic map is an excellent example of bifurcation and the period-doubling route to chaos, as well as the concept of sensitive dependence on initial conditions (SDIC). It reveals how even the simplest non-linear system can create great complexity. Used as a basic population model, the logistic equation is just the product of linear growth, X, and linear decay, (1 - X), scaled by a driving factor, alpha. The map is the iteration X(i+1) = alpha * X(i) * (1 - X(i)). For small alpha (which controls the rate of birth and death), the population reaches a stable, steady state value - i.e. birth rate and death rate have attained equilibrium. Eventually as alpha is increased the population suddenly starts to oscillate between two stable values, regardless of the initial number. As alpha is increased further still, each stable value itself bifurcates (splits into two, period-doubling) to give overall four stable values, then eight, sixteen etc. until there are infinite - chaos! This happens at an ever increasing rate, which turns out to be governed by a newly discovered universal constant known as the Feigenbaum constant, d = 4.6692.... In words, the next period doubling happens over a distance in alpha a factor of 4.6692 sooner than the previous period doubling. One implication of this is that once period-doubling starts, it exponentially accelerates to chaos (in the driving parameter space). Such behaviour has also been observed in a range of physical systems, for example convection cells in a closed volume, transition from laminar to turbulent flow, dripping faucets.... It is very easy to hear the difference between the regions of chaos (fuzzy in appearance) and the ordered regions which can re-appear momentarily, despite the system having already entered a chaotic state. Within the chaotic regions there exists a sensitivity to initial conditions. What this means is that if you changed the initial value of the system by even the tiniest amount, for the same number of iterations you will get a very different result! This is NOT due to randomness, but rather the complexity of the system. This chaos is actually deterministic, meaning that it can be exactly repeated / computed at any time given only the rules of the system and the initial conditions. It is important to consider this in reality, just because something appears too complex to describe does not mean it is random. There may be a surprisingly simple set of rules behind it all, and it just seems complex because the rules have played out long enough and involved a large enough number of entities. Whether this applies to the entire Universe or not is a question many deep thinkers have pondered on since the dawn of time, as the consequences are profound when it comes to concepts such as free will and the existence of God.... Coded in LabVIEW (still images and audio rendering). Video put together using Hitfilm Express. This particular output used a sampling rate of 5760 Hz and 192 samples per alpha (to give 30 fps matching with the video frame-rate). Alpha was incremented such that it represented one pixel at 1080p resolution. This led to a decent length video with a frequency not unbearably high or inaudibly low!