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Extra info for Chaotic Dynamics - An Introduction Based on Classical Mechanics
23. Tank with two outlets. The outlets, when opened alternately, generate chaotic advection in a flat container. (a) and (b) illustrate the flow in the first and second half period, respectively. The flow itself is very simple; the advection of the particles is nevertheless chaotic. in a flowing medium (air or water). The environmental significance of this matter is obvious. Consider a large and flat container with two point-like outlets. Water whirls while flowing out. The two outlets are alternately open, each for half a period (Fig.
1. 17 18 Fig. 24. Paths of two dye particles (continuous and broken lines) starting near each other in a tank with two outlets (situated at (−1, 0) and (1, 0)). The consecutive black dots (squares) indicate the instants when the left (right) outlet is opened. In the initial instant the left outlet is opened. The time spent in the tank is very different for the two particles. The phenomenon: complex motion, unusual geometry 2 y 1 0 –1 –2 –2 –1 0 1 2 x Fig. 25. Shape of a dye drop initially and after five periods in the tank with two outlets.
3) only, not the exponent of the power law, the dimension. 11 What is the algorithm given in terms of the removal of pieces from a unit cube which leads to the direct product of three Cantor sets of parameter r = 1/3 placed along the edges of a unit cube? 7(a)) at the same resolution, ε = 3−n ? 3 Thin and fat fractals Covering a fractal embedded in d dimensions with d-dimensional cubes of size ε, the observed volume V (ε) of a fractal is given by V (ε) = εd N (ε). 3) of the number of covering boxes, the 6 The observed surface area in this generalised concept is S(ε) ∼ εd−1 N (ε), which is for d = 2, for example, the observed perimeter length.