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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1. The basin of attraction for an attracting set is strictly confined within for any initial value and for any parameter values and .

2. The basin of attraction for an attracting set abruptly vanishes at for any value of , that is, "all boundary crises occur at ."

3. Since the function is symmetric around , this map is particularly convenient for renormalization group analysis.

The blue box on the left is the locator where the image inside the box is rescaled on the right in accordance with the zoom-in level . By dragging the locator, you can change where to zoom in.

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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.



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