Trajectory-Scaling Functions for Generalized Logistic Maps with z-Unimodality

Requires a Wolfram Notebook System
Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.
This Demonstration shows Feigenbaum's trajectory scaling function (TSF) [1–6],
[more]
Contributed by: Ki-Jung Moon (January 2014)
Based on a program by: Stephen Wolfram
Open content licensed under CC BY-NC-SA
Snapshots
Details
1. The test map used in this Demonstration, , generalizes the well-known logistic map
[7–14]. Here
;
is the subcontrol parameter (which determines the unimodality, the degree of the local maximum of
).
2. The program for this Demonstration presents a large collection of super-stable parameter values for period-doubling periodic attractors within
and
, which were obtained using a high-precision Newton algorithm with fixed precision 50. These values are accurate up to 45 decimal places, so you can use them for your own research or study.
3. For , the lower bound of the domain of
approaches
(i.e.
) and
approaches the universal scaling function
(i.e.
) satisfying the following properties:
universality: ,
,
,
,
symmetry: for
,
for
and
,
where is the second Feigenbaum constant as a function of
[1–6,15].
4. For the practical purpose of numerical simulation, is good enough to observe the universality [1, 3]. In 1988, Andrew L. Belmonte et al. used
(i.e.
cycles obtained from the logistic map) for comparison with their experimental data [4].
References
[1] M. J. Feigenbaum, "The Transition to Aperiodic Behavior in Turbulent Systems," Communications in Mathematical Physics, 77, 1980 pp. 65–86.
[2] M. J. Feigenbaum, "The Metric Universal Properties of Period Doubling Bifurcations and the Spectrum for a Route to Turbulence," Annals of the New York Academy of Sciences, 357, 1980 pp. 330–336.
[3] M. J. Feigenbaum, "Universal Behavior in Nonlinear Systems," Los Alamos Sciences, 1, 1980 pp. 4–27.
[4] A. L. Belmonte, M. J. Vinson, J. A. Glazier, G. H. Gunaratne, and B. G. Kenny, "Trajectory Scaling Functions at the Onset of Chaos: Experimental Results," Physical Review Letters, 61(5), 1988 pp. 539–542.
[5] M. C. de S. Vieira and G. H. Gunaratne, "The Trajectory Scaling Function for Period Doubling Bifurcations in Flows," Journal of Statistical Physics, 58, 1990 pp. 1245–1256.
[6] E. Mayoral and A. Robledo, "Tsallis' Index and Mori's
Phase Transitions at the Edge of Chaos," Physical Review E, 72(2), 2005 p. 026209.
[7] S. H. Strogatz, Nonlinear Dynamics and Chaos, New York: Perseus Books Publishing, 1994.
[8] K. T. Alligood, T. D. Sauer, and J. A. Yorke, Chaos: An Introduction to Dynamical Systems, New York: Springer, 1996.
[9] H.-O. Peitgen, H. Jürgens, and D. Saupe, Chaos and Fractals: New Frontiers of Science, 2nd ed., New York: Springer, 2004.
[10] S. Wolfram, A New Kind of Science, Champaign, IL: Wolfram Media, 2002.
[11] M. J. Feigenbaum, "Quantitative Universality for a Class of Non-Linear Transformations," Journal of Statistical Physics, 19, 1978 pp. 25–52.
[12] M. J. Feigenbaum, "The Universal Metric Properties of Nonlinear Transformations," Journal of Statistical Physics, 21, 1979 pp. 669–706.
[13] K.-J. Moon, "Reducible Expansions and Related Sharp Crossovers in Feigenbaum's Renormalization Field," Chaos: An Interdisciplinary Journal of Nonlinear Science, 18, 2008 p. 023104.
[14] K.-J. Moon, "Erratum: Reducible Expansions and Related Sharp Crossovers in Feigenbaum's Renormalization Field," Chaos: An Interdisciplinary Journal of Nonlinear Science, 20, 2010 p. 049902.
[15] K. Briggs, "Feigenbaum Scaling in Discrete Dynamical Systems," Ph.D. thesis, Department of Mathematics, Melbourne University, Australia, 1997. keithbriggs.info/documents/Keith_Briggs_PhD.pdf.
Permanent Citation