# Bifurcation Diagram for a Generalized Logistic Map

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This Demonstration shows a bifurcation diagram for a generalized logistic map, [1–7]. This map is very well-suited for numerical analysis because:

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Contributed by: Ki-Jung Moon (October 2013)

Based on programs by: Ed Pegg Jr, Enrique Zeleny, C. Pellicer-Lostao, and R. Lopez-Ruiz

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

• The test map generalizes the well-known logistic map [1–5].

• The initial condition is fixed at .

• For or for , the iterates of rapidly approach (more rapidly for larger values of and ) and therefore, due to the finite length in precision, the numbers are too large to compute [8]. A little trick is used to avoid this problem.

References

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

[4] M. J. Feigenbaum, "Quantitative Universality for a Class of Nonlinear Transformations," *Journal of Statistical Physics*, 19(1), 1978 pp. 25–52. doi:10.1007/BF01020332.

[5] M. J. Feigenbaum, "The Universal Metric Properties of Nonlinear Transformations," *Journal of Statistical Physics*, 21(6), 1979 pp. 669–706. doi:10.1007/BF01107909.

[6] K.-J. Moon and S. D. Choi, "Reducible Expansions and Related Sharp Crossovers in Feigenbaum's Renormalization Field," *Chaos: An Interdisciplinary Journal of Nonlinear Science*, 18(2), 2008 pp. 023104. doi:10.1063/1.2902826.

[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. doi:10.1063/1.3530128.

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

## Permanent Citation