Logistic map

The logistic map is an archetypical example of how very complex, chaotic behaviour can arise from very simple non-linear dynamical equations. The map was popularized by the biologist Robert May in 1976. It was originally made as a very simple model for the population numbers of species in the presence of limiting factors such as food supply or disease, containing two causal loops: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 current population
• due to starvation, the population will increase at a rate proportional to the value obtained by taking the theoretical "carrying capacity" of the environment less the current population.

Mathematically this can be written as

x_n+1 = r x_n (1 - x_n),

where:

x_n is a number between zero and one, and represents the population at year n

x₀ is a number between zero and one, and represents the initial population (at year 0)

r is a positive number, and represents a combined rate for reproduction and starvation.

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 population.
• With r between 1 and 2, the population will quickly stabilize on a single value; this value depends on r but does not depend on the initial population.
• With r between 2 and 3, the population will also eventually stabilize on a single value, but first oscillates around that value for some time. Again, the final value does not depend on the initial population
• With r between 3 and 1+√6 (approximately 3.45), the population will oscillate between two values forever. These two values are dependent on r but independent of the initial population.
• 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 population.
• 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 population.
• At r = 3.57 (approximately) is the onset of chaos. We can no longer see any oscillations. Slight variations in the initial population yield dramatically different results over time, a prime characteristic of chaos.
• Most values beyond 3.57 exhibit chaotic behaviour, but there are still certain isolated of r that appear to show non-chaotic behavior; 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.
• Beyond r = 4, the values eventually leave the interval [0,1] and diverge for almost all initial values.

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.

A 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