WOLFRAM NOTEBOOK

Feigenbaum's Scaling Relation for Superstable Parameter Values: "Bifurcation Diagram Helper"

bifurcation order, r

As the fixed points of the iterated function

i+1

=f(

)=λ

(1-

)

approach chaotic behavior with repeated period doubling, the superstable parameter value

approaches its limiting value

∞

, known as the Feigenbaum point. As the value of

(the order of the period-doubling bifurcation) increases, Feigenbaum's scaling relation between the three adjacent superstable parameter values approaches a universal limit

δ=4.6692016…

known as Feigenbaum's constant[1–7], that is,

lim

r∞

r+1

r+2

=δ=4.6692016…

References

[1] K. T. Alligood, T. D. Sauer, and J. A. Yorke, Chaos: An Introduction to Dynamical Systems, New York: Springer, 1996.

[2] S. H. Strogatz, Nonlinear Dynamics and Chaos, New York: Perseus Books Publishing, 1994.

[3] S. Wolfram, A New Kind of Science, Champaign, IL: Wolfram Media, 2002.

[4] M. J. Feigenbaum, "Quantitative Universality for a Class of Non-Linear Transformations," Journal of Statistical Physics, 19, 1978 pp. 25–52.

[5] M. J. Feigenbaum, "The Universal Metric Properties of Nonlinear Transformations," Journal of Statistical Physics, 21, 1979 pp. 669–706.

[6] K.-J. Moon, "Reducible Expansions and Related Sharp Crossovers in Feigenbaum's Renormalization Field," Chaos: An Interdisciplinary Journal of Nonlinear Science, 18, 2008 pp. 023104.

[7] K.-J. Moon, "Erratum: Reducible Expansions and Related Sharp Crossovers in Feigenbaum's Renormalization Field," Chaos: An Interdisciplinary Journal of Nonlinear Science, 20, 2010 pp. 049902.