Details

1. The test map used in this Demonstration, , generalizes the well-known logistic map [7–14]. Here ; is the subcontrol parameter (which determines the unimodality, the degree of the local maximum of ).

2. The program for this Demonstration presents a large collection of super-stable parameter values for period-doubling periodic attractors within and , which were obtained using a high-precision Newton algorithm with fixed precision 50. These values are accurate up to 45 decimal places, so you can use them for your own research or study.

3. For , the lower bound of the domain of approaches (i.e. ) and approaches the universal scaling function (i.e. ) satisfying the following properties:

universality: , , , ,

symmetry: for , for and ,

where is the second Feigenbaum constant as a function of [1–6,15].

4. For the practical purpose of numerical simulation, is good enough to observe the universality [1, 3]. In 1988, Andrew L. Belmonte et al. used (i.e. cycles obtained from the logistic map) for comparison with their experimental data [4].

References

[1] M. J. Feigenbaum, "The Transition to Aperiodic Behavior in Turbulent Systems," Communications in Mathematical Physics, 77, 1980 pp. 65–86.

[2] M. J. Feigenbaum, "The Metric Universal Properties of Period Doubling Bifurcations and the Spectrum for a Route to Turbulence," Annals of the New York Academy of Sciences, 357, 1980 pp. 330–336.

[3] M. J. Feigenbaum, "Universal Behavior in Nonlinear Systems," Los Alamos Sciences, 1, 1980 pp. 4–27.

[4] A. L. Belmonte, M. J. Vinson, J. A. Glazier, G. H. Gunaratne, and B. G. Kenny, "Trajectory Scaling Functions at the Onset of Chaos: Experimental Results," Physical Review Letters, 61(5), 1988 pp. 539–542.

[5] M. C. de S. Vieira and G. H. Gunaratne, "The Trajectory Scaling Function for Period Doubling Bifurcations in Flows," Journal of Statistical Physics, 58, 1990 pp. 1245–1256.

[6] E. Mayoral and A. Robledo, "Tsallis' Index and Mori's Phase Transitions at the Edge of Chaos," Physical Review E, 72(2), 2005 p. 026209.

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

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

[9] H.-O. Peitgen, H. Jürgens, and D. Saupe, Chaos and Fractals: New Frontiers of Science, 2nd ed., New York: Springer, 2004.

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

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

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

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

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

[15] K. Briggs, "Feigenbaum Scaling in Discrete Dynamical Systems," Ph.D. thesis, Department of Mathematics, Melbourne University, Australia, 1997. keithbriggs.info/documents/Keith_Briggs_PhD.pdf.