Journal of Applied Science and Engineering

Published by Tamkang University Press


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Chyi-Lung Lin This email address is being protected from spambots. You need JavaScript enabled to view it.1 and Mon-Ling Shei1

1Department of Physics, Soochow University, Taipei, Taiwan 111, R.O.C.


Received: May 12, 2005
Accepted: March 21, 2006
Publication Date: March 1, 2007

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We prove that the well-known logistic map, f(x) = μx(1 x), is topologically conjugate to the map f(x) = (2 - μ) x(1 - x). The logistic map thus has the same dynamics at parameter values μ and 2 - μ, and hence has the μ → 2 - μ symmetry in dynamics. To examine this symmetry, we study the (μ, s)n relation of fn , which is obtained by eliminating x from the equations fn (x) = x and s = dfn (x)/dx. We then obtain an equation directly relating μ and s for period-n point of f. We derive the (μ, s)n relation for period n = 1, 2, 3, and 4, and we show that the (μ, s)n relations are invariant under the transformation of μ → 2 - μ.

Keywords: Logistic Map, Topologically Conjugate, Symmetry, Invariant, Periodic Bifurcation


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