What is Chaos?: Dynamical Instabilities

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	Having understood what is meant by determinism, initial conditions, and uncertainty of measurements, you can now learn about dynamical instability, which to most physicists is the same in meaning as chaos.
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	Dynamical instability refers to a special kind of behavior in time found in certain physical systems and discovered around the year 1900, by the physicist Henri Poincaré.
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	Poincaré was a physicist interested in the mathematical equations which describe the motion of planets around the sun.
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	The equations of motion for planets are an application of Newton's laws, and therefore completely deterministic.
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	That these mathematical orbit equations are deterministic means, of course, that by knowing the initial conditions---in this case, the positions and velocities of the planets at a given starting time---you find out the positions and speeds of the planets at any time in the future or past.
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	Dynamical Instabilities [6/20]	Home
	Of course, it is impossible to actually measure the initial positions and speeds of the planets to infinite precision, even using perfect measuring instruments, since it is impossible to record any measurement to infinite precision. Thus there always exists an imprecision, however small, in all astronomical predictions made by the equation forms of Newton's laws.
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	Dynamical Instabilities [7/20]	Home
	Up until the time of Poincaré, the lack of infinite precision in astronomical predictions was considered a minor problem, however, because of a tacit assumption made by almost all physicists at that time.
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	Dynamical Instabilities [8/20]	Home
	The assumption was that if you could shrink the uncertainty in the initial conditions---perhaps by using finer measuring instruments---then any imprecision in the prediction would shrink in the same way.
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	Dynamical Instabilities [9/20]	Home
	In other words, by putting more precise information into Newton's laws, you got more precise output for any later or earlier time. Thus it was assumed that it was theoretically possible to obtain nearly-perfect predictions for the behavior of any physical system.
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	But Poincaré noticed that certain astronomical systems did not seem to obey the rule that shrinking the initial conditions always shrank the final prediction in a corresponding way.
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	By examining the mathematical equations, he found that although certain simple astronomical systems did indeed obey the "shrink-shrink" rule for initial conditions and final predictions, other systems did not.
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	Dynamical Instabilities [12/20]	Home
	The astronomical systems which did not obey the rule typically consisted of three or more astronomical bodies with interaction between all three. For these types of systems, Poincaré showed that a very tiny imprecision in the initial conditions would grow in time at an enormous rate.
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	Thus two nearly-indistinguishable sets of initial conditions for the same system would result in two final predictions which differed vastly from each other.
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	Poincaré mathematically proved that this "blowing up" of tiny uncertainties in the initial conditions into enormous uncertainties in the final predictions remained even if the initial uncertainties were shrunk to smallest imaginable size.
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	Dynamical Instabilities [15/20]	Home
	That is, for these systems, even if you could specify the initial measurements to a hundred times or a million times the precision, etc., the uncertainty for later or earlier times would not shrink, but remain huge.
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	The gist of Poincaré's mathematical analysis was a proof that for these "complex systems," the only way to obtain predictions with any degree of accuracy at all would entail specifying the initial conditions to absolutely infinite precision.
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	Dynamical Instabilities [17/20]	Home
	For these astronomical systems, any imprecision at all, no matter how small, would result after a short period of time in an uncertainty in the deterministic prediction which was hardly any smaller than if the prediction had been made by random chance.
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	Dynamical Instabilities [18/20]	Home
	The extreme "sensitivity to initial conditions" mathematically present in the systems studied by Poincaré has come to be called dynamical instability, or simply chaos.
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	Dynamical Instabilities [19/20]	Home
	Because long-term mathematical predictions made for chaotic systems are no more accurate that random chance, the equations of motion can yield only short-term predictions with any degree of accuracy.
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	Dynamical Instabilities [20/20]	Home
	Although Poincaré's work was considered important by some other foresighted physicists of the time, many decades would pass before the implications of his discoveries were realized by the science community as a whole.. One reason was that much of the community of physicists was involved in making new discoveries in the new branch of physics called quantum mechanics, which is physics extended to the atomic realm.
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