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:
“At this very moment,... the most frightful horrors are taking place in every corner of the world. People are being crushed, slashed, disembowelled, mangled; their dead bodies rot and their eyes decay with the rest. Screams of pain and fear go pulsing through the air at the rate of eleven hundred feet per second. After travelling for three seconds they are perfectly inaudible. These are distressing facts; but do we enjoy life any the less because of them? Most certainly we do not.”
—Aldous Huxley (18941963)
“At this very moment,... the most frightful horrors are taking place in every corner of the world. People are being crushed, slashed, disembowelled, mangled; their dead bodies rot and their eyes decay with the rest. Screams of pain and fear go pulsing through the air at the rate of eleven hundred feet per second. After travelling for three seconds they are perfectly inaudible. These are distressing facts; but do we enjoy life any the less because of them? Most certainly we do not.”
—Aldous Huxley (18941963)
“I have all my life been on my guard against the information conveyed by the sense of hearingit being one of my earliest observations, the universal inclination of humankind is to be led by the ears, and I am sometimes apt to imagine that they are given to men as they are to pitchers, purposely that they may be carried about by them.”
—Mary Wortley, Lady Montagu (16891762)
“I have never worked for fame or praise, and shall not feel their loss as I otherwise would. I have never for a moment lost sight of the humble life I was born to, its small environments, and the consequently little right I had to expect much of myself, and shall have the less to censure, or upbraid myself with for the failures I must see myself make.”
—Clara Barton (18211912)
“In the true sense ones native land, with its background of tradition, early impressions, reminiscences and other things dear to one, is not enough to make sensitive human beings feel at home.”
—Emma Goldman (18691940)
“As, therefore, we can have no dependence upon morality without religion;Mso, on the other hand, there is nothing better to be expected from religion without morality;Mnevertheless, tis no prodigy to see a man whose real moral character stands very low, who yet entertains the highest notion of himself, in the light of a religious man.”
—Laurence Sterne (17131768)
“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 (18391897)
“What is Americanism? Every one has a different answer. Some people say it is never to submit to the dictation of a King. Others say Americanism is the pride of liberty and the defence of an insult to the flag with their gore. When some half-developed person tramples on that flag, we should be ready to pour out the blood of the nation, they say. But do we not sit in silence when that flag waves over living conditions which should be an insult to all patriotism?”
—Anna Howard Shaw (18471919)