# High-Precision Newton Algorithm for Generalized Logistic Maps with Unimodality *z*

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This Demonstration shows a table of superstable parameter values of a period-doubling periodic attractor. The test map is defined as , which generalizes the well-known logistic map . Here is an iteration number, , and is the unimodality (or the degree) of the local maximum of . The superstable parameter values are used for the renormalization group analysis of many low-dimensional dynamical systems with chaotic behavior. See the references [1–4].

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Contributed by: Ki-Jung Moon (November 2013)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Snapshot 1 for ; snapshot 2 for ; snapshot 3 for ; snapshot 4 for ; snapshot 5 for . The generalized logistic map is particularly convenient for renormalization group analysis because:

1. The attracting domain (or the basin of attraction) of any periodic/chaotic attractor is strictly confined within for any and .

2. All boundary crises occur at for any .

Mitchell J. Feigenbaum's original renormalization group analysis on the classic logistic map can be found in [1] and [2]. More information on the renormalization group analysis can be found in [3] and [4].

References

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

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

[3] 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.

[4] 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.

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