Physics 303, Fall 2014 1

Logistic Map, Feignbaum number and Lyapunov exponentPhysics 303, University of New Mexico

M. Gold

1 Logistic map

Generating the map: an amazing demohttp://en.wikipedia.org/wiki/File:LogisticCobwebChaos.gif!

Figure 1: “cobweb” plot

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Figure 2: Logistic map bifurcation diagrams, from http://mathworld.wolfram.com/

LogisticMap.html

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Figure 3: Enlarged section of Logistic map bifurcation diagrams, fromhttp://mathworld.wolfram.com/LogisticMap.html. Period doubling ends atr = 3.5699456720.

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Table 1: sequence of bifurcation points for period doubling cascade. (period doublingvalues from Chaos: An Introduction to Dynamical Systems, Kathleen T. Alligood) Be-yond value of r=3.5699456720 period doubling ends.

n rn ∆rn δn1 3.00000002 3.4494897 0.4494897 4.75144663 3.5440903 0.0946006 4.65622884 3.5644073 0.0203170 4.66832105 3.5687594 0.0043521 4.66863336 3.5696916 0.00093227 3.5698913 0.00019978 3.5699340 0.0000427∞ 3.5699456720 4.6692016091...

Feigenbaum number is limit of δn in the following sequence. Let rn be the n-thbifurcation point,

∆rn = rn − rn−1 and δn = ∆rn/∆rn+1 Feigenbaum number is universal for period-doubling cascade.

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Figure 4: Feigenbaum’s calculation used lots of precision!

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Feignbaum, Los Alamos Science, 1980

Figure 5: Feigenbaum’s function g1(x)

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2 Quantifying Chaos: Lyapunov exponent

Λ = limn→∞

1

n

n−1∑i=0

ln

∣∣∣∣ dfdx |xi

∣∣∣∣

Lyapunov Exponents for the Logistic Map

zoomhorizontal

vertical

reset

detail0.75 0.80 0.85 0.90 0.95 1.00

0.2

0.4

0.6

0.8

1.0Bifurcation Diagram

0.75 0.80 0.85 0.90 0.95 1.000

Lyapunov Exponent

!

This Demonstration plots the orbit diagram of the logistic map xk+1 = 4 l xk H1 - xkL and the corresponding Lyapunov exponents for different ranges of the parameter l.

The Lyapunov exponent is a parameter characterizing the behavior of a dynamical system. It gives the average rate of exponential divergence from nearby initial conditions. The Lyapunov exponent of the logistic map is given by 1

n ⁄k=1n log @4 l H1 - 2 xkLD.

If the Lyapunov exponent is positive, then the system is chaotic; if it is negative, the system will converge to a periodic state; and if it is zero, there is a bifurcation.

By dragging the locator to the left or right or clicking the plot, you can scroll through the whole range of l-values (0.70–1.0), generate the bifurcation diagram, and plot the Lyapunov exponent over that range.

You can zoom to the position of the locator by using the zoom sliders. To replot the graphs at higher zoom scales, use the "detail" button to increase the number of l values and the number of iterations to 5000.

Figure 6: Lyapunov exponent for logistics map, Demo from Wolfram. “Forcing” valueon horizontal axis is our r/4. A value of λ > 0 implies chaos. For super stable fixedpoints, λ→∞

http://demonstrations.wolfram.com/LyapunovExponentsForTheLogisticMap/

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3 What does this have to do with the DDP?

Figure 7: Commercially built Daedalon chaotic pendulum used by Davidson http:

//www.reed.edu/physics/faculty/illing/campus/pdf/DavidsonThesis12.pdf

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Figure 8: Example phase-space from DDP http://www.reed.edu/physics/faculty/

illing/campus/pdf/DavidsonThesis12.pdf

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Figure 9: Bifurcation diagram from DDP data. http://www.reed.edu/physics/

faculty/illing/campus/pdf/DavidsonThesis12.pdf

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4 Fractal Dimension

Hausdorf dimension d of set in p-dimensional space: N(ε) = number of p-cubes of sideε needed to cover set.

d =lim

ε→ 0

(lnN(ε)

ln 1/ε

)Cantor set: start with unit length line, iteratively remove center third.

Figure 10: Cantor set http://en.wikipedia.org/wiki/List_of_fractals_by_

Hausdorff_dimension

iteration n covering N = 2n ε = (1/3)n

0 1 1 11 1/3 + 1/3 2 1/32 1/9 + 1/9 + 1/9 + 1/9 4 1/9

Table 2: iterating the Cantor set. dimension p=1, so “cubes” in this case are just lines

d = limn→∞

ln 2n

ln 3n=

ln 2

ln 3= 0.6309...

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Figure 11: Calculated fractal dimension of logistics map = 0.538 http://en.

wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension

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Figure 12: “The Dimension of the Planar Brownian Frontier is 4/3” http://arxiv.

org/abs/math/0010165 Math.Res.Lett.8:401-411,2001

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Figure 13: Identifying cancer with fractal dimensional analysis http://www.

sciencedaily.com/releases/2011/04/110418093852.htm “Fractal analysis of im-ages of breast tissue specimens provides a numeric description of tumor growth patternsas a continuous number between 1 and 2. This number, the fractal dimension, is anobjective and reproducible measure of the complexity of the tissue architecture of thebiopsy specimen. The higher the number, the more abnormal the tissue is.”

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Figure 14: By C. Fukushima and J. Westerweel, Technical University of Delft, TheNetherlands. This picture is a false-color image of the far-field of a submerged turbulentjet, made visible by means of laser induced fluorescence

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5 Fractals in Art

Figure 15: Mount Fuji Seen Below a Wave at Kanagawa, Hokusai

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Figure 16: Autumn Rhythm, Jackson Pollack

Fractal? Specifically, can a true Pollack be authenticated by a fractal analysis?

• Yes, Richard P. Taylor, Adam P. Micolich and David Jonas (1999) https://

plus.maths.org/content/fractal-expressionism https://cpb-us-e1.wpmucdn.

com/blogs.uoregon.edu/dist/e/12535/files/2015/12/PollockPRL-1gq9u97.

pdf

• No, “Drip Paintings and Fractal Analysis” Katherine Jones-Smith, Harsh Mathur,Lawrence M. Krauss, Phys. Rev. E 79, 046111 (2009);

“Revisiting Pollock’s Drip Paintings,” Katherine Jones-Smith and Harsh Mathur,Nature 444, E9 (2006).”

• Yes, “Authenticating Pollock paintings using fractal geometry”, RP Taylor, RGuzman, TP Martin, GDR Hall, Pattern Recognition Letters, 2007;

“Perceptual and physiological responses to Jackson Pollock’s fractals”, R Taylor,B Spehar, C Hagerhall, P Van Donkelaar, Frontiers in human neuroscience, 2011;

“Order in Pollock’s chaos”, RP Taylor, Scientific American, 2002

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6 Fractals in Biology

Figure 17: Example of fractal in biology

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7 Scaling in Biology

Figure 18: Galileo’s original drawing, showing how larger animals’ bones must begreater in diameter compared to their lengths. d2 ∝ L3

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Figure 19: ”Scaling of Skeletal Mass to Body Mass in Birds and Mammals”, HenryD. Prange, John F. Anderson and Hermann Rahn The American Naturalist Vol. 113,No. 1 (Jan., 1979), pp. 103-122

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• “Scale and Dinension”, LANL preprint Los Alamos Science Summer/Fall 1984:http://panda3.phys.unm.edu/nmcpp/gold/phys303/west-lanl-preprint.pdf

• “ Chaos and fractals in human physiology,” A. L. Goldberger, D. R. Rigney, B.J. West, Sci Am 1990;262:40-49.

• “ A General Model for the Origin of Allometric Scaling Laws in Biology Au-thor(s): Geoffrey B. West, James H. Brown and Brian J. Enquist ,” Science,New Series, Vol. 276, No. 5309 (Apr. 4, 1997), pp. 122-126.

• “The Fourth Dimension of Life: Fractal Geometry and Allometric Scaling ofOrganisms,” Geoffrey B. West [1,2], James H. Brown [3] and Brian J. Enquist [3]Science, New Series, Vol. 284, No. 5420 (Jun. 4, 1999), pp. 1677-1679.

• “The origin of allometric scaling laws in biology from genomes to ecosystems:towards a quantitative unifying theory of biological structure and organization,”Geoffrey B. West [1,2] and James H. Brown [1,3], The Journal of ExperimentalBiology 208, 1575-1592.

1. The Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA.

2. Theoretical Division, MS B285, Los Alamos National Laboratory, Los Alamos,NM 87545, USA.

3. Department of Biology, University of New Mexico, Albuquerque, NM 87131,USA.

Sante Fe Institute Resources:

• http://www.santafe.edu/research/themes/physics-and-computation-complex-systems

• http://www.complexityexplorer.org

Abstract, from “The Fourth Dimension of Life” http://www.sciencemag.org/

content/284/5420/1677.abstract

Fractal-like networks effectively endow life with an additional fourth spatial dimen-sion. This is the origin of quarter-power scaling that is so pervasive in biology. Or-ganisms have evolved hierarchical branching networks that terminate in size-invariantunits, such as capillaries, leaves, mitochondria, and oxidase molecules. Natural selec-tion has tended to maximize both metabolic capacity, by maximizing the scaling ofexchange surface areas, and internal efficiency, by minimizing the scaling of transportdistances and times. These design principles are independent of detailed dynamics andexplicit models and should apply to virtually all organisms.

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Figure 20: From http://hep.ucsb.edu/courses/ph6b_99/0111299sci-scaling.

html

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Figure 21: From http://hep.ucsb.edu/courses/ph6b_99/0111299sci-scaling.

html

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Quoting again, from “The Fourth Dimension of Life:”

We have proposed that the quarter-power allometric scaling laws and otherfeatures of the dynamical behaviour of biological systems reflect the constraintsinherent in the generic properties of these networks. These were postulated tobe: (i) networks are space-filling in order to service all local biologicallyactive subunits; (ii) the terminal units of the network are invariants; and(iii) performance of the network is maximized by minimizing the energyand other quantities required for resource distribution.

These properties of the ‘average idealized organism’ are presumed to beconsequences of natural selection. Thus, the terminal units of the network whereenergy and resources are exchanged (e.g. leaves, capillaries, cells, mitochondria orchloroplasts), are not reconfigured or rescaled as individuals grow from newborn toadult or as new species evolve. In an analogous fashion, buildings are supplied bybranching networks that terminate in invariant terminal units, such as electricaloutlets or water faucets. The third postulate assumes that the continuous feedbackand fine-tuning implicit in natural selection led to ‘optimized’ systems. For example,of the infinitude of space-filling circulatory systems with invariant terminal units thatcould have evolved, those that have survived the process of natural selection,minimize cardiac output. Such minimization principles are very powerful, becausethey lead to ‘equations of motion’ for network dynamics.

Nota Bene– There is a crucial physics difference between pulsatile and non-pulsatile systems.

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Figure 22: “space filling fractal”, From Geoffrey B. West, James H. Brown and BrianJ. Enquist Science, New Series, Vol. 276, No. 5309 (Apr. 4, 1997), pp. 122-126

Figure 23: From “The Fourth Dimension of Life”, Geoffrey B. West, James H. Brownand Brian J. Enquist (1999)

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Figure 24: From “The Origin of Allometric Scaling:”

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Figure 25: From “The Origin of Allometric Scaling:”

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Figure 26: From “The Origin of Allometric Scaling:”

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Figure 27: From “The Origin of Allometric Scaling:”