Finite Lyapunov Exponent for Generalized Logistic Maps with z-Unimodality

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram CDF Player or other Wolfram Language products.

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

This Demonstration shows a finite Lyapunov exponent of a one-dimensional unimodal map , which is a generalization of the well-known logistic map . The related Lyapunov exponent can be defined by

[more]

,

where is the base-2 logarithm, is the iteration number, is the iterate of starting from the initial condition , is the main control parameter, and is the subcontrol parameter, which determines the unimodality, the degree of the local maximum of .

[less]

Contributed by: Ki-Jung Moon (January 2014)
Based on a program by: Stephen Wolfram
Open content licensed under CC BY-NC-SA


Snapshots


Details

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. Jürgens, and D. Saupe, Chaos and Fractals: New Frontiers of Science, 2nd ed., New York: Springer, 2004.

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

[5] M. J. Feigenbaum, "Quantitative Universality for a Class of Non-Linear 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 pp. 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 pp. 049902.



Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.
Send