Sensitivity to Intial conditions

Sensitivity to Initial Conditions: An extremely small change in the initial conditions of a chaotic or non-linear system leads to extremely differing results. Any arbitrarily small interval of initial values will be enlarged significantly by iteration. This is the so-called "butterfly effect" in which the flapping wings of a single butterfly could theoretically make the difference whether or not a hurricane occurred in another place and time. (The title of a paper by Edward N. Lorentz was "Can the flap of a butterfly's wing stir up a tornado in Texas?" 

Lorentz discovered sensitivity to initial conditions by trying to repeat a computer modelling of the weather with a very slightly rounded off number. The results rapidly diverged from the previous calculation. Sensitivity to intitial conditions is an indication of error propagation. It can also be understood as the generation of information: there were two distinct starting points even if we couldn't distinguish between them, and running the system has generated that information. 

The " clinamen" or swerve is the classical account of chance fluctuations leading to the structures of the world. 

K-flows, named after the Russian mathematician Andrei Kolmogorov are a measure of chaos. K- entropy is a measure of the average rate at which trajectories starting from points extremely close together are moving apart. In a chaotic system an initial deviation will soon become as large as the true "signal" itself. Calculators or computers that round off numbers, to no matter how many digits, rapidly lead to errors in equations that both expand the number of digits and are sensitive to infinitesimally small differences in the numbers. However, in these cases K is positive but not infinite. It is infinite in totally random paths.

While sensitivity is central to chaos, it does not automatically lead to chaos. For example the simple function x » cx (c>1) is a linear transformation and does not lead to chaos, but any deviation is magnified during the course of iteration. given an initial error e, the error after n iterations will be c to nth power times e.