Accepted:
00/00/1994
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/vol01/luzeau01/ | © Copyright 1994 | |||
| Volume 1 | Received: Accepted: |
00/00/1994 00/00/1994 |
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From beta-expansions to chaos and fractals
Luzeaux, D. |
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| Abstract | |
| In this paper we will study the behavior of the following difference equation: xk+1 = {Bxk}, where B may be an integer, a real, or even a complex or a matrix, and {.} is the fractional part; depending on B and x0, the nature of the solution xk changes dramatically and chaotic behavior occurs; by studying different properties of the set of all xk, we look for eventual cycles, density results, and discuss Lebesgue measure, category and fractal dimension problems. This equation occurs in applied mathematics, for instance in control theory when one considers a single input linear time-invariant system controlled by a particular state feedback control. Well-known fractal sets can also be deduced from that equation for particular values of B. Many questions related to this difference equation are still open problems in number theory and in spite of its apparent simplicity, this equation turns out to be a real challenge. | |
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Multimedia Links (none) Reference Links (none) Citation Reference Luzeaux, D. (1994), From beta-expansions to chaos and fractals, Complexity International, Volume 1, Paper ID: luzeau01, URL: http://www.complexity.org.au/vol01/luzeau01/ |
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