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:
“All of us failed to match our dreams of perfection. So I rate us on the basis of our splendid failure to do the impossible.”
—William Faulkner (1897–1962)
“I believe it has been said that one copy of The Times contains more useful information than the whole of the historical works of Thucydides.”
—Richard Cobden (1804–1865)
“Perhaps it is a universal truth that the loss of liberty at home is to be charged to provisions against danger, real or pretended, from abroad.”
—James Madison (1751–1836)
“I can understand that if you have sold arms to the ayatollah why you might not be quite as sensitive to the need to get assault weapons off our streets.”
—Charles S. Robb (b. 1939)
“This immediate dependence of language upon nature, this conversion of an outward phenomenon into a type of somewhat in human life, never loses its power to affect us. It is this which gives that piquancy to the conversation of a strong-natured farmer or backwoodsman, which all men relish.”
—Ralph Waldo Emerson (1803–1882)
“For those parents from lower-class and minority communities ... [who] have had minimal experience in negotiating dominant, external institutions or have had negative and hostile contact with social service agencies, their initial approaches to the school are often overwhelming and difficult. Not only does the school feel like an alien environment with incomprehensible norms and structures, but the families often do not feel entitled to make demands or force disagreements.”
—Sara Lawrence Lightfoot (20th century)
“As long as fathers rule but do not nurture, as long as mothers nurture but do not rule, the conditions favoring the development of father-daughter incest will prevail.”
—Judith Lewis Herman (b. 1942)