Movement of double pendulum
Let’s model the population when a = 2. If we pick a starting value of, say p =
0.3 at time = 0, then to find the population at time = 1, we feed p = 0.3 into a
× p × (1 – p) to give 0.42. Using only a handheld calculator we can repeat this
operation, this time with p = 0.42, to give us the next figure (0.4872).
Progressing in this way, we find the population at later times. In this case, the
population quickly settles down to p = 0.5. This settling down always takes place
for values of a less than 3.
If we now choose a = 3.9, a value near the maximum permissible, and use
the same initial population p = 0.3 the population does not settle down but
oscillates wildly. This is because the value of a is in the ‘chaotic region’, that is, a
is a number greater than 3.57. Moreover, if we choose a different initial
population, p = 0.29, a value close to 0.3, the population growth shadows the
previous growth pattern for the first few steps but then starts to diverge from it
completely. This is the behaviour experienced by Edward Lorenz in 1961 (see
box).
Population changing over time for a = 3.9