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

As the fixed points of the iterated function 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 known as Feigenbaum's constant [1–7], that is,

[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.