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The logistic map is a formula used in the 19th century to simulate population growth with limited computational resources. It is a deterministic simulation that follows the equation f(x) = f(x-1)r(1-f(x-1), or more commonly, xₙ₊₁ = xₙr(1-xₙ). r is a number from 1 to 4 inclusively, and x is the ratio from the current population the the maximum population. When r is less than 1, the population inevitably expreiences extinction; where x goes towards 0. When r is between 1 and 3 inclusively, x goes towards a fixed point. Mathmatically, lim n->Infinity xₙ=a, where a is a fixed point. When x is between 3 and 3.56995 exclusively, x starts oscillating between 2^b numbers where b is a natural number. When x is over 3.56995, there is chaos. In a situation named chaos theory or the butterfly effect, tiny changes to x can cause completely different results. However, oscillation orders not a power of 2 can arise. The Logistic Map is now a popular learning example to explain chaos theory, a phenomenon where the smallest changes can affect results. Even floating-point errors can have such effects. Visit https://en.wikipedia.org/wiki/Logistic_map for more information.
