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Page 81
... periodic orbit , a set consisting of the points x , T ( x ) , ... , T " -1 ( x ) , and T ” ( x ) = x . Invariant sets are further classified according to how points near the invariant set behave under T. An invariant set M is called an ...
... periodic orbit , a set consisting of the points x , T ( x ) , ... , T " -1 ( x ) , and T ” ( x ) = x . Invariant sets are further classified according to how points near the invariant set behave under T. An invariant set M is called an ...
Page 82
... periodic orbit of order 2. It turns out that there is a natural invariant set under the iterates of T that lies in the interval [ 0,1 ] . To find it we consider successive iterations of T and keep track of the parts of the interval [ 0 ...
... periodic orbit of order 2. It turns out that there is a natural invariant set under the iterates of T that lies in the interval [ 0,1 ] . To find it we consider successive iterations of T and keep track of the parts of the interval [ 0 ...
Page 89
... periodic points . This is an interesting possibility . ) Third , it is possible to bootstrap oneself up from generating homeomorphisms of the interval to generating homeomorphisms of the square , the cube , and so on . These ...
... periodic points . This is an interesting possibility . ) Third , it is possible to bootstrap oneself up from generating homeomorphisms of the interval to generating homeomorphisms of the square , the cube , and so on . These ...
Page 96
... periodic limit set of order 3 ( that is , its limit set consisted of three internal points traversed in a certain order ) , four had internal periods of order 2 , one showed no limiting behavior at all , and one converged to an internal ...
... periodic limit set of order 3 ( that is , its limit set consisted of three internal points traversed in a certain order ) , four had internal periods of order 2 , one showed no limiting behavior at all , and one converged to an internal ...
Page 98
... periodic limit sets ( some of quite high order ) . Of most interest to us , however , were 334 transformations that exhibited no periodic limiting behavior , suggesting that their limit sets contained infinite numbers of points . Some ...
... periodic limit sets ( some of quite high order ) . Of most interest to us , however , were 334 transformations that exhibited no periodic limiting behavior , suggesting that their limit sets contained infinite numbers of points . Some ...
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Popular passages
Page 299 - Talibus orabat dictis, arasque tenebat, cum sic orsa loqui vates : ' Sate sanguine divom, 125 Tros Anchisiade, facilis descensus Averno ; noctes atque dies patet atri janua Ditis ; sed revocare gradum superasque evadere ad auras, hoc opus, hie labor est.
Page 231 - But it is not always so; it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible, and we have the fortuitous phenomenon.
Page 210 - The low-velocity equation of state for a lattice gas can be written as p - ^p (l - |v2), where p is the pressure, p is the density, and v is the flow speed.
Page 316 - Southeastern Conference on Combinatorics. Graph Theory, and Computing (Florida Atlantic University, Boca Raton, Florida, April 2-6, 1979), 3-18.
Page 121 - He is a member of the National Academy of Sciences and a Fellow of the American Academy of Arts and Sciences, the American Physical Society and the American Association for the Advancement of Science.
Page 231 - A very small cause which escapes our notice determines a considerable effect that we cannot fail to see, and then we say that the effect is due to chance. If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of that same universe at a succeeding moment. But even if it were the case that the natural laws...
Page 224 - Instead, we conclude by remarking that really efficient high-speed computing devices may, in the field of non-linear partial differential equations as well as in many other fields, which are now difficult or entirely denied of access, provide us with those heuristic hints which are needed in all parts of mathematics for genuine progress.
Page 272 - Let us say here that the results of our computations show features which were, from the beginning, surprising to us. Instead of a gradual, continuous flow of energy from the first mode to the higher modes, all of the problems show an entirely different behavior.
Page 58 - That was dramatized by Bertrand, who considered the problem of finding the probability that a chord of a circle chosen "at random" be longer than the side of an equilateral triangle inscribed in the circle.