| | | | | | 2.1.3— A Simple Equation Can Produce Complicated Output | | | | | | | | | The phenomenon of chaos is surprising. | | | | | | | | | Let's follow the deterministic equation, step by step, to see how it produces a set of data that looks random: | | | | | | | | | The first value of x at n=1 was chosen to be x(1) = .892. | | | | | | | | | The value x(n+l) at the (n+l)-th step is computed from the previous value x(n) at the n-th step by using the equation that x(n+1) = 3.95 x(n) [1 - x(n)]. | | | | | | | | | Thus, to compute the next value of x at n=2, multiply 3.95 by .892 and then multiply that result by (1-.892). This yields x(2) = .380. | | | | | | | | | To compute the next value of x at n=3, multiply 3.95 by .380 and then multiply that result by (1-.380). This yields x(3) = .931. | | | | | | | | | Continue the same process. | | | | | | | | | This deterministic mechanism alone produces a seemingly random sequence of values x(n). That is why the discoverers of this phenomenon called it chaos. | | | | |