A separatrix is a phase path separating bounded from unbounded motion, and corresponds to a object oscillating in unstable equibrium. For example, on a swing a childs motion is usually bounded (back and forth) but with enough force he can swing all the way over and around, which is unbounded motion. If his trajectory were on a separatrix, he would go back and forth but he would reach the highest point possible, directly over the bar. That is indeed a very unstable equilibrium; it presupposes a balance of kinetic and potential energy so precarious, that the child does not drop directly on his head, though such danger would appear very likely to manifest itself. As Marion and Thornton stated, Motion in the vicinity of such a separatrix is extremely sensitive to initial conditions because points on either side of the separatrix have very different trajectories. The particular types of oscillation indicated here, pendular and rotational, are not an exhaustive representation of the class. Now, certain discontinuous oscillations are illustrative of sensitivity to initial conditions. The logistic equation is representative in this respect. For certain coefficients it rapidly converges to a single result; for others, the result oscillates between either two or four possible values; for still others, it produces wildly varying results, a potentially infinite number of them, though all lie between 0 and 1. Although the logistic equation represents a pattern, one might say, under certain conditions its results have no apparent rhyme or reason. The specific and technical term for this phenomenon is chaos. A characteristic of chaotic systems is that after a certain number of iterations it is impossible to predict their solutions, unless one has a model perfectly describing the phenomenon in question. With sufficient scrutiny and measurement, it might be possible to form such a model for simple systems like double pendula; even then, it is unlikely that the model would offer perfect accuracy for a truly indefinite duration. It was and remains an immense achievement that men discovered equations of nature, but that they might discover chaotic equations accurately describing phenomena in nature is a dubious view of the extent of our intellectual powers. . . .
Posted on: Sat, 15 Mar 2014 05:15:37 +0000
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