Logistic map

The logistic map is an archetypical example of how very complex, chaotical behaviour can arise from very simple non-linear dynamical equations. The map was popularized by biologist Robert May in 1976. The map was originally made as a very simple model for the population numbers of species in the presence of limiting factors such as food supply or disease, containing two causal loops:

due to reproduction the population will increase at a rate proportional to the population
due to limiting factors, the population will decrease at a rate proportional to the size of the population (e.g. if their are many foxes in one year, they will eat most of the rabbits, and the next generation of foxes will be small since there is no food).

Mathematically this can be written as

x[n + 1] = r * x[n] * (1 - x[n]),

where x[n] is the population at year n (normalized to be between 0 and 1) and r is the combined rate for reproduction and starvation, chosen to be between 0 and 4 to ensure that x[n+1] is again a number between 0 and 1.

By varying the parameter r, the following behaviour is observed:

With r between 0 and 1, the population will eventually die, independent of the initial value x[0].
With r between 1 and 3, the population will quickly stabilize on a single value; this value depends on r but does not depend on the the initial value x[0].
With r between 3 and 1+√6 (approximately 3.45), the population will oscillate between two values forever. These two values are again dependent on r but independent of the initial x[0].
With r between 3.45 and 3.54 (approximately), the population will oscillate between four values forever; again, this behavior does not depend on the initial value.
With r slightly bigger than 3.54, the population will oscillate between 8 values, then 16, 32, etc. The lengths of the parameter intervals which yield the same number of oscillations decrease rapidly; the ratio between the lengths of two successive such bifurcation intervals approaches the Feigenbaum constant δ = 4.669... All of these behaviors do not depend on the initial value.
At r = 3.57 (approximately) is the onset of chaos. We can no longer see any oscillations. Slightly varying the starting value x[0] of the equation yields dramatically different results over time, a prime characteristic of chaos.
Most values beyond 3.57 exhibit chaotic behaviour, but there are still certain isolated non-chaotic ranges of r; for instance around 3.82 there is a range of parameters r which show oscillation between three values, and for slightly higher values of r oscillation between 6 values, then 12 etc. There are other ranges which yield oscillation between 5 values etc.; all oscillation periods do occur. These behaviours are again independent of the initial value.

A bifurcation diagram summarizes this. The horizontal axis shows the values of the parameter r while the vertical axis shows the possible long-term values of x.

The bifurcation diagram is a fractal: if you zoom in on the above mentioned value r = 3.82 and focus on one arm of the three, say, the situation nearby looks just like a shrunk and slightly distorted version of the whole diagram. The same is true for all other non-chaotic points. This is an example of the deep and ubiquitous connection between chaos and fractals.

An GNU Octave script to generate bifurcation diagrams can be found at Logistic_map/Computer simulation.

External links:

Dan Marthaler: The Logistic Map, http://mathpost.la.asu.edu/~daniel/logistic.html . Contains an interactive computer simulation of the logistic map and also allows to zoom in on the bifurcation diagram.
Another interactive simulation: http://www.geocities.com/CapeCanaveral/Hangar/7959/logisticmap.html