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 of, rate, information, loss, sensitive, dependence, initial and/or conditions:
“We all run on two clocks. One is the outside clock, which ticks away our decades and brings us ceaselessly to the dry season. The other is the inside clock, where you are your own timekeeper and determine your own chronology, your own internal weather and your own rate of living. Sometimes the inner clock runs itself out long before the outer one, and you see a dead man going through the motions of living.”
—Max Lerner (b. 1902)
“We all run on two clocks. One is the outside clock, which ticks away our decades and brings us ceaselessly to the dry season. The other is the inside clock, where you are your own timekeeper and determine your own chronology, your own internal weather and your own rate of living. Sometimes the inner clock runs itself out long before the outer one, and you see a dead man going through the motions of living.”
—Max Lerner (b. 1902)
“Knowledge is of two kinds. We know a subject ourselves, or we know where we can find information upon it.”
—Samuel Johnson (1709–1784)
“This loss of interest, hair, and enterprise ...”
—Philip Larkin (1922–1986)
“Too much traffic with a quotation book begets a conviction of ignorance in a sensitive reader. Not only is there a mass of quotable stuff he never quotes, but an even vaster realm of which he has never heard.”
—Robertson Davies (b. 1913)
“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)
“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)
“The Supreme Court would have pleased me more if they had concerned themselves about enforcing the compulsory education provisions for Negroes in the South as is done for white children. The next ten years would be better spent in appointing truant officers and looking after conditions in the homes from which the children come. Use to the limit what we already have.”
—Zora Neale Hurston (1891–1960)