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Time-Series Analysis for Generalized Logistic Maps with z-Unimodality

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This Demonstration shows a time-series plot of an iterative map with a barcode attachment. The test map, , generalizes the well-known logistic map [1–7]. Here is the iteration number, is the iterate of starting from the initial value , is the main control parameter, and is the subcontrol parameter (which determines the unimodality, the degree of the local maximum of ).

Contributed by: Ki-Jung Moon (December 2013)
Based on a program by: Stephen Wolfram
Open content licensed under CC BY-NC-SA

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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] 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.

[5] 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.

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

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

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