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

《淡江理工學刊》 10卷1期 (2007/03) Pp. 89-94

並列摘要

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 f^n, which is obtained by eliminating x from the equations f^n(x)=x and s=df^n(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.

並列關鍵字

Logistic Map ； Topologically Conjugate ； Symmetry ； Invariant ； Periodic Bifurcation