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Extra resources for Chaotic Dynamics - An Introduction Based on Classical Mechanics
22. Number of collisions of 20 000 particles starting with unit velocity at right angles to the line segment drawn in Fig. 21, as a function of the y co-ordinate. ) from which an arbitrary number of bounces can, in principle, occur (the particles then become trapped among the discs), but these do not form an interval: they form a scattered fractal cloud along the line segment. Three or four Christmas-tree ornaments in contact with each other reflect light several times before light reaches our eyes.
It can be shown that advection then corresponds to dissipative systems. The advection dynamics can then have attractors, often even chaotic ones. This implies that pollutant particles accumulate along a fractal pattern on the surface of the fluid. This phenomenon can indeed be observed in lakes, bays and harbours as a direct consequence of chaos! 3 Phase space Our examples have shown that the traditional representation via the displacement or velocity vs. time graphs does not provide a suitable overview of the motion, since, however long the observation time may be, one can always expect some further novel behaviour.
The surface-tovolume ratio is therefore finite and becomes smaller as the size becomes larger. 1 On the other hand, it is known that there exist macroscopic objects with large surface area. These are always porous, with ramified or pitted surfaces. Effective chemical catalysts, for example, must have a large surface. The need for rapid gas exchange accounts for the large surfaceto-volume ratio of the respiratory organs. The surface area of the human lungs (measured at microscopic resolution), for example, is the same as 1 This is used for example in thermodynamics, when the internal energy of a system is considered to be the sum of the internal energies of the finite volume elements, thus neglecting the interactions of the volume elements across the surfaces.
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