Mitchell Feigenbaum's one-term parameter scaling laws for the logistic map are

and ,

where (1) is the order of the period-doubling pitchfork bifurcation, (2) is the control parameter of the logistic map, (3) is the superstable parameter value for each bifurcation order, (4) is a special parameter value corresponding to the Feigenbaum point of the logistic map, (5) and are constants, (6) is the Feigenbaum constant.

Superstable parameter values can be obtained using the standard Newton iterative scheme. For more information, please see the references in the "Details" section.

More information on Feigenbaum's one-term parameter scaling laws can be found in [1–4].

References

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

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

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

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