Chyi-Lung Lin 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
Download Citation:
||https://doi.org/10.6180/jase.2007.10.1.10
ABSTRACT
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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