Logistic Map

The logistic map is a discrete dynamical system defined by the equation x sub n+1 equals r times x sub n times the quantity one minus x sub n, a simple rule that repeatedly feeds its own output back in as the next input. It grew out of Pierre Francois Verhulst's earlier logistic differential equation for population growth, and was popularized as a discrete demographic model by the biologist Robert May in a widely read 1976 paper. As the parameter r is increased, the map's long-term behavior moves from settling at a fixed value, to oscillating between a small number of values through a series of period-doubling splits, to fully chaotic behavior in which tiny differences in starting value produce wildly different outcomes. The ratio at which each period-doubling step occurs approaches a fixed value called the Feigenbaum constant, and chaos sets in around r approximately equal to 3.56995, making the logistic map one of the simplest equations known to produce chaotic dynamics.