This Demonstration shows the iterate of the test map , for superstable parameter values of perioddoubling periodic attractors as a function of . The test map generalizes the wellknown logistic map, . Here is the iteration number, is the iterate of starting from the initial condition (i.e. ), is the control parameter, is the bifurcation order (e.g. for , for , for , etc.), is the superstable parameter value for each bifurcation order, and is the subcontrol parameter (which determines the unimodality, the degree of the local maximum of ).
The program for this Demonstration presents a large collection of within the unimodality interval and the bifurcation order interval , which were obtained using a highprecision Newton algorithm with fixed precision 50. These values are accurate to 45 decimal places, so you can use them for your own study or research. To imitate the special iterates at the edge of chaos ( ), you can use instead of for any . [1] S. H. Strogatz, Nonlinear Dynamics and Chaos, New York: Perseus Books Publishing, 1994. [2] K. T. Alligood, T. D. Sauer, and J. A. Yorke, Chaos: An Introduction to Dynamical Systems, New York: Springer, 1996. [3] S. Wolfram, A New Kind of Science, Champaign, IL: Wolfram Media, 2002. [4] H.O. Peitgen, H. Jürgens, and D. Saupe, Chaos and Fractals: New Frontiers of Science, 2nd ed., New York: Springer, 2004. [5] M. J. Feigenbaum, "Quantitative Universality for a Class of NonLinear Transformations," Journal of Statistical Physics, 19, 1978 pp. 25–52. [6] M. J. Feigenbaum, "The Universal Metric Properties of Nonlinear Transformations," Journal of Statistical Physics, 21, 1979 pp. 669–706. [7] 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. [8] 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.
