The Logistic map is an archetypical example of how very complex behaviour can arise from very simple non-linear dynamical equations. The map was originally made as a very simple model for predator-prey systems like foxes and rabbits, containing two casual loops:
- due to reproduction the population will increase at a rate proportional to the population
- due to limited resources (e.g. prey), the population will decrease at a rate proportional to the size of the population (e.g. if the foxes eat all the rabbits one year, there will be no food the next year and all foxes will die from starvation).
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 the be between 0 and 1) and r is the combined rate for reproduction and starvation.
By simulating this system with varying r, we can observe the following behaviour:
- With r between 0 and 1, the population will die.
- With r between 1 and 2, the population will quickly stabilize on a single value.
- With r between 2 and 3, the population will oscillate between two values but will ultimately converge to a single value.
- With r is between 3 and 3.4 (approximately), the population will oscillate between two values.
- With r between 3.4 and 3.6, the possible values of the oscillations within the population start to double faster and faster as r increases.
- At r = 3.7 (approximately) is the onset of chaos. We can no longer see any oscillations. Varying the starting value of the equations yields dramatically different results over time.
- At r = 4 the values exhibit true chaotic behaviour.
An interactive simulation of the Logistic map can be found at http://mathpost.la.asu.edu/~daniel/logistic.html.