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:
““Terence, this is stupid stuff:
You eat your victuals fast enough;
There can’t be much amiss, ‘tis clear,
To see the rate you drink your beer.”
—A.E. (Alfred Edward)
“Phenomenal nature shadows him wherever he goes. Clouds in the staring sky transmit to one another, by means of slow signs, incredibly detailed information regarding him. His inmost thoughts are discussed at nightfall, in manual alphabet, by darkly gesticulating trees. Pebbles or stains or sunflecks form patterns representing in some awful way messages which he must intercept. Everything is a cipher and of everything he is the theme.”
—Vladimir Nabokov (1899–1977)
“Our ego ideal is precious to us because it repairs a loss of our earlier childhood, the loss of our image of self as perfect and whole, the loss of a major portion of our infantile, limitless, ain’t-I-wonderful narcissism which we had to give up in the face of compelling reality. Modified and reshaped into ethical goals and moral standards and a vision of what at our finest we might be, our dream of perfection lives on—our lost narcissism lives on—in our ego ideal.”
—Judith Viorst (20th century)
“Peasants are a rude lot, and hard: life has hardened their hearts, but they are thick and awkward only in appearance; you have to know them. No one is more sensitive to what gives man the right to call himself a man: good-heartedness, bravery and virile brotherhood.”
—Jacques Roumain (1907–1945)
“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)
“No punishment has ever possessed enough power of deterrence to prevent the commission of crimes. On the contrary, whatever the punishment, once a specific crime has appeared for the first time, its reappearance is more likely than its initial emergence could ever have been.”
—Hannah Arendt (1906–1975)
“There must be a world revolution which puts an end to all materialistic conditions hindering woman from performing her natural role in life and driving her to carry out man’s duties in order to be equal in rights.”
—Muammar Qaddafi (b. 1938)