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Additional resources for Chaotic Dynamics - An Introduction Based on Classical Mechanics
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. 24 2 Fractal objects that of a tennis court (approximately 100 m2 ), while the volume is only a few litres (10−3 m3 ).
In the first step the centre third of the unit interval is removed, but afterward it is 1/9 (and not 1/3) of the remaining 1/3 intervals, then 1/27 of the new segments, etc. At the nth step, 1/3n of the 2 Fractal objects 39 Fig. 11. Fat Cantor set. In contrast to a Cantor set, a decreasing proportion of the segments is removed. These proportions are 1/3, 1/9, 1/27, . . , 1/3n . The total length of the fat Cantor set constructed in this manner converges to a finite number greater than zero. The dimension is D0 = 1 and the fat fractal exponent, α, is 0 < α < 1.
In the investigation of the statistical properties of chaos, a major role was played by, among others, B. Chirikov, M. Berry, L. Bunimovich, J. P. Eckmann, H. Fujisaka, P. Grassberger, C. Grebogi, M. H´enon, P. Holmes, L. Kadanoff, E. Ott, O. R¨ossler, D. Ruelle, Y. Sinai, and S. Smale. 3); this has been pioneered by H. Aref, P. Cvitanovi´c, J. Gollub, A. Libchaber, R. May, C. Nicolis, H. Swinney, Y. Ueda, J. Wisdom, and others. 1 What is a fractal? 1 Objects with large surfaces It is taken for granted that the surface or volume of a traditional geometrical object, for example a sphere or a cube, is well defined.