The Sound of the Logistic Map

In case you ever wondered what the logistic map sounded like! The logistic map is an excellent example of bifurcation and the period-doubling route to chaos, as well as the concept of sensitive dependence on initial conditions (SDIC). It reveals how even the simplest non-linear system can create great complexity. Used as a basic population model, the logistic equation is just the product of linear growth, X, and linear decay, (1 - X), scaled by a driving factor, alpha. The map is the iteration X(i 1) = alpha * X(i) * (1 - X(i)). For small alpha (which controls the rate of birth and death), the population reaches a stable, steady state value - i.e. birth rate and death rate have attained equilibrium. Eventually as alpha is increased the population suddenly starts to oscillate between two stable values, regardless of the initial number. As alpha is increased further still, each stable value itself bifurcates (splits into two, period-doubling) to give overall four stable values, then eight, sixteen et