Logistic Map f(x) = µx(1 -x) is Topologically Conjugate to the Map f(x) = (2 - µ) x(1 - x)

Chyi-Lung Lin; Mon-Ling  Shei

doi:10.6180/jase.2007.10.1.10

Logistic Map f(x) = µx(1 -x) is Topologically Conjugate to the Map f(x) = (2 - µ) x(1 - x)

Physics

The period doubling bifurcation diagram for the map f(x) = µx(1 – x), where µ is extended to the range µ 0.

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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