Monte Carlo simulation of bifurcations in the logistic map

Points per r-value:

Maximum point age:

Minimum r-value:

Frame rate:

A_n

The logistic map was originally used to approximate an animal population over time. Mathematically, it is written as:

A_n+1 = rA_n(1 - A_n)

where:

• A_n represents the population at year n, with 0 <= A_n <= 1. Of course, in reality a population is a whole number, but this makes the maths simpler. A value of 1 means the maximum population i.e. the capacity of the physical environment.
• r represents a combined rate for reproduction and starvation, with r > 0.

All initial populations A₀ eventually settle into one of three behaviours:

1. A fixed value.
2. Periodic oscillation between values.
3. Chaotic i.e. unpredictable values.

A bifurcation diagram shows these three behaviours. The x-axis shows r and the y-axis shows the population. The population eventually stabilises to a fixed value for r = 1 to 3, and you can see the periodic behavior starting after this, eventually becoming chaotic at around r = 3.5699457...

The Monte Carlo simulation on the left uses repeated random sampling to produce the bifurcation diagram. The algorithm is roughly:

1. For each value of r, generate a multiple points with a random initial population and random maximum age (remaining number of iterations).
2. Set the population of each point to A_n+1 = rA_n(1-A_n) and update its remaining number of iterations.
3. If the remaining number of iterations for any point reaches zero, then we pick a new random initial population and age for that point.
4. Clear the plotting area and plot each point at (r, A_n).
5. Go to 2.

The plot is updated for every iteration, so you can see how the populations are changing over time.