Rate of Information Loss and Sensitive Dependence On Initial Conditions
One hallmark of chaotic dynamics is the loss of information as simulation occurs. If we start with information on the first s bits of the initial iterate, then after m simulated iterations (m<s) we only have (s-m) bits of information remaining. Thus we lose information at the exponential rate of one bit per iteration. After s iterations, our simulation has reached the fixed point zero, regardless of the true iterate values; thus we have suffered a complete loss of information. This illustrates sensitive dependence on initial conditions—the mapping from the truncated initial condition has deviated exponentially from the mapping from the true initial condition. And since our simulation has reached a fixed point, for almost all initial conditions it will not describe the dynamics in the qualitatively correct way as chaotic.
Equivalent to the concept of information loss is the concept of information gain. In practice some real-world process may generate a sequence of values {} over time, but we may only be able to observe these values in truncated form. Suppose for example that = .1001101, but we only observe the truncated value .1001 . Our prediction for is .001 . If we wait until the real-world process has generated the true value .001101, we will be able to observe the truncated value .0011, which is more accurate than our predicted value .001 . So we have received an information gain of one bit.
Read more about this topic: Dyadic Transformation
Famous quotes containing the words rate, information, loss, sensitive, dependence, initial and/or conditions:
“You are more than entitled not to know what the word ‘performative’ means. It is a new word and an ugly word, and perhaps it does not mean anything very much. But at any rate there is one thing in its favor, it is not a profound word.”
—J.L. (John Langshaw)
“When action grows unprofitable, gather information; when information grows unprofitable, sleep.”
—Ursula K. Le Guin (b. 1929)
“No performance is worth loss of geniality. ‘Tis a cruel price we pay for certain fancy goods called fine arts and philosophy.”
—Ralph Waldo Emerson (1803–1882)
“Pity in a man of knowledge seems almost laughable, like sensitive hands on a cyclops.”
—Friedrich Nietzsche (1844–1900)
“The invalid is a parasite on society. In a certain state it is indecent to go on living. To vegetate on in cowardly dependence on physicians and medicaments after the meaning of life, the right to life, has been lost ought to entail the profound contempt of society.”
—Friedrich Nietzsche (1844–1900)
“Capital is a result of labor, and is used by labor to assist it in further production. Labor is the active and initial force, and labor is therefore the employer of capital.”
—Henry George (1839–1897)
“People nowadays have such high hopes of America and the political conditions obtaining there that one might say the desires, at least the secret desires, of all enlightened Europeans are deflected to the west, like our magnetic needles.”
—G.C. (Georg Christoph)