Chesapeake Bay Mathematics Undergraduate Colloquium University of Utah 09/03/2014 Fractals in Nature Ivan Sudakov www.math.utah.edu/~sudakov www.mathclimate.org
References 1. W. Szemplinska-Stupnicka. Chaos Bifurcations and Fractals Around Us: a Brief Introduction. World Scientific, 2003. 2. K. Falconer. Fractals. A Very Short Introduction. Oxford University Press, 2013 3. D.P. Feldman. Chaos and Fractals: An Elementary Introduction. Oxford University Press, 2012 4. M. Frantz & A. Crannell. Viewpoints: Mathematical Perspective and Fractal Geometry in Art. Princeton University Press, 2011 5. M. Novak et al. Thinking in Patterns. Fractals and Related Phenomena in Nature. World Scientific, 2004. 6. B. Mandelbrot. The Fractal Geometry of Nature. Freeman and Company, 1982. 7. M. F. Barnsley. Fractals Everywhere. Academic Press,1994. 8. S. Lovejoy. Area-Perimeter Relation for Rain and Cloud Areas // Science, 216 (4542), 185-187,1982. 9. J.R. Krummel et al. Landscape Patterns in a Disturbed Environment // Oikos, 48, 321-324, 1987. 10. C. Hohenegger et al. Transition in the Fractal Geometry of Arctic Melt Ponds // Cryosphere, 6 (5), 1157-1162, 2012.
The Logistic System X next
Population Growth and the Gypsy Moth
Population Growth and the Gypsy Moth X next Next years population rate of growth = r X this years population
Population Growth and the Gypsy Moth X next Next years population rate of growth = r X this years population Human population growth curve
Population Growth and the Gypsy Moth Positive feedback Negative feedback X next = r X (1-X) The logistic function or logistic curve models the S-curve of growth of some set P. The initial stage of growth is approximately exponential; then, as competition arises, the growth slows, and at maturity, growth stops. Equilibrium state
Modeling an Evolutionary System X next : A Model of Deterministic Chaos (A.k.a. the Logistic or Verhulst Equation) X next = rx (1-X) X = population size - expressed as a fraction between 0 (extinction) and 1 (greatest conceivable population) X next is what happens at the next iteration or calculation of the equation. Or, it is the next generation r = rate of growth - that can be set higher or lower. It is the positive feedback. It is the tuning knob (1-X) acts like a regulator (the negative feedback), it keeps the growth within bounds since as X rises, 1-X falls
Modeling an Evolutionary System X next : A Model of Deterministic Chaos (A.k.a. the Logistic or Verhulst Equation) X next = rx (1-X) Logistic population ranges between 0 (extinction) and 1 (highest conceivable population) Iterated algorithm is calculated over and over Recursive the output of the last calculation is used as the basis of the next calculation
Value of X (Populaton size) 1.0 0.5 0.0.02 X next and Deterministic Chaos.13.05.05.02.65.64.58.60.61.58.35 Modeling an Evolutionary System.62 X next = rx (1-X) r = 2.7 Equilibrium state X =.02 and r = 2.7 X next = rx (1-X) X next = (2.7) (.02) (1-.02 =.98) X next =.0529 Number of Equation Iterations.62 Iteration X Value 0 0.0200000 1 0.0529200 2 0.1353226 3 0.3159280 4 0.5835173 5 0.6561671 6 0.6091519 7 0.6428318 8 0.6199175 9 0.6361734 10 0.6249333 11 0.6328575 12 0.6273420 13 0.6312168 14 0.6285118 15 0.6304087 16 0.6290826 17 0.6300117 18 0.6293618 44 0.6296296 45 0.6296296 46 0.6296296 47 0.6296296 48 0.6296296 49 0.6296296 50 0.6296296
But, what about these irregularities? Are they just meaningless noise, or do they mean something? Last run at 20 generations
Experimenting With X next and Deterministic Chaos X next = rx (1-X) A time-series diagram
r = 2.7
r = 2.9
r = 3.0
r = 3.1
r = 3.3
r = 3.4
r = 3.5
r = 3.6
r = 3.7
r = 3.8
r = 3.9
r = 4.0
r = 4.1
Learning Outcomes 1. Computational Viewpoint In a dynamic system the only way to know the outcome of an algorithm is to actually calculate it; there is no shorter description of its behavior. 2. Positive/Negative Feedback Behavior stems from the interplay of positive and negative feedbacks. 3. r Values Rate of Growth, or how hard the system is being pushed. High r means the system is dissipating lots of energy and/or information. 4. Deterministic does not equal Predictable At high r values the behavior of the system becomes inherently unpredictable.
Converting a Time Series Diagram Into a Bifurcation Diagram
Population Size = X This axis was a time series, but becomes... r Value r = 2.9 r = 2.7 X =.629 X =.655
It would be reasonable for population sizes in the gap to fall between those for r = 2.7 and r = 2.9 And for population size to drop as r drops r = 2.9 r = 2.7 X =.629 X =.655
split split split r = 3.3 r = 3.5 X =.48 &.82 X =.50,.87,.38,.82
r = 3.8 0.877682831619863 0.407951579058487 0.917802935168261 0.286674687986186 0.777070782765993 0.658280769082272 0.854799352927153 0.471646192817398 0.946945034149357 0.190912518507075 0.58696672938057 0.921259794327218 0.275652705596884 0.758739507677205 0.695604695234438 0.804607452168521 0.597414340316927 0.913939695942348 0.298884926867995 0.796300363964611 0.616383158394868
Population Size Modeling an Evolutionary System Bifurcation Diagram A Bifurcation is a change in basic behavior of a system Very Simple Behavior 1 st Bifurcation 3 rd Bifurcation 2 nd Bifurcation Very Complex Behavior r Values Rate of Growth
Population Size Modeling an Evolutionary System Bifurcation Diagram Why are some systems stable, reliable, predictable, and others are not? 1 st Bifurcation 3 rd Bifurcation 2 nd Bifurcation Refrigerators Computers Cars Airplanes Weather Climate Stock Market Human Behavior Because we engineer human-made systems to operate at low r r Values Rate of Growth
Learning Outcomes 5. Bifurcations Bifurcations are a change in the behavior of the system, the entire range of behaviors or a particular system can be shown in one bifurcation diagram. 6. Instability increases with r The harder the system is pushed, the higher the r value the more unstable and unpredictable its behavior becomes. This is seen in the bifurcation cascade: settling to one population value, splitting to 2, then 4, then 8, etc. population values, and finally visiting so many population values a pattern cannot be seen.
Self- Similarity: Fractals
Universality Properties of Complex Evolutionary Systems Fractal Organization - X next patterns, within patterns, within patterns 1 Full Bifur cat ion Diagr am 2 Red box in 1 Stretched and Enlarged in 2 window opens on magnificat ion
Universality Properties of Complex Evolutionary Systems Fractal Organization - X next patterns, within patterns, within patterns 2 3 window opens on magnificat ion Red box in 2 Stretched and Enlarged in 3 window opens on magnificat ion
Universality Properties of Complex Evolutionary Systems Fractal Organization - X next patterns, within patterns, within patterns 3 4 window opens on magnificat ion Red box in 3 Stretched and Enlarged in 4 sm all r ed box is enlar ged on anot her page, and opens anot her window
Universality Properties of Complex Evolutionary Systems Fractal Organization - X next The closer we zoom in the patterns, within patterns, within patterns 4 more the detail we see, and Red box in 4 Stretched and Enlarged here. we see similar patterns repeated again and again. sm all r ed box is enlar ged on anot her page, and opens anot her window
This is Self Similarity Similarities at all scales of observation Patterns, within patterns, within patterns FRACTAL
Euclidean and Fractal Geometry Things that are fractal are characterized by two distinctive Dimension 0 characteristics: Dimension 1 Dimension 2 3 1. Non-whole Dimensions Fractal Dimension = log N (number of similar pieces) log M (magnification factor) N M D Hexahedron Tet rahedron Oct ahedr on Dodecahedr on Icosahedron Dimension 0 Dimension 1 Dimension 2 Dimension 3 N = # of new pieces M = magnification D = dimension Fractal dimensions are never whole numbers D log log N M
Euclidean and Fractal Geometry Things that are fractal are characterized by two distinctive characteristics: 2. Generated by iteration Fractal objects are generated by iteration of an algorithm, or formula. The Koch Curve is an example, generated by 4 steps, which are then repeatediterated -over and over indefinitely, or as long as you want. Koch Curve First Iteration 1. Begin with a line 2. Divide line into thirds 3. Remove middle portion 4. Add two lines to form a triangle in middle third of original line Repeat Steps 1-4
Universality Fractal Geometry Koch Curve 2 nd Iteration 3 rd Iteration 4 th Iteration 5 th Iteration
Koch Curve Fractal Dimensions 2 3 1 4 D = 1 2 3 log N (number of new pieces) log 4 0.602 log M (magnification: factor of finer resolution) = = log 3 0.477 Koch's Curve has a dimension of 1.2618595071429
What you can see and understand... Depends on Your Scale of Observation
Fractal Temperature Patterns in Time 1,000 Year Record 20,000 Year Record zoomed to.. 20,000 Year Record
Fractal Temperature Patterns in Time 20,000 Year Record zoomed to.. 20,000 Year Record 450,000 Year Record
Universality Properties of Complex Evolutionary Systems Fractal Organization Sea Level Changes
Scale and Observation What you can measure depends on the scale of your ruler. The time you can resolve depends on the accuracy of your clock. The size of what you can see depends on the power of your measuring instrument; microscopes for small things, eyes, for intermediate things, telescopes for very distant things. The Earth events you can witness, or even the human species can witness, depends on how long you live. There is no typical or average size for events.
How Long is the Coast of Great Britain? It depends on the length of your ruler The red ruler measures a longer coastline. http://en.wikipedia.org/wiki/list_of_fractals_by_hausdorff_dimension
How Long is the Coast of Great Britain? It depends on the length of your ruler The coast line is actually infinitely long Fractal Dimension = 1.24 http://en.wikipedia.org/wiki/list_of_fractals_by_hausdorff_dimension
Fractional dimension for the Eastern Shore of the Chesapeake Bay is 1.46
Procedure: Measure the projected cloud area A and the perimeter P of each cloud Define a linear size through l A Perimeter dimension define through: P ~ l D For ordinary Euclidean objects: log P Slope: D= 1 logl
Pioneered by Lovejoy (Science 1982) Area-perimeter analysis of projected cloud patterns using satellite and radar data Suggest a perimeter dimension D=4/3 ( 1.3) of projected clouds Instead of Consequences: Cloud perimeter is fractal and hence selfsimilar in a non-trivial way Makes it possible to ascribe a (quantitative) number that characterizes the structure Slope 4/3
Lena River Delta D=1.58 Yakut Permafrost Lakes D=1.84
(a) Fractal dimension (D) of forest patches in the vicinity of Natchez, Mississippi, as a function of patch size. (b) Section of the original map illustrating how small patches tend to be simple in shape. (c) Section of the original map illustrating the more complex shapes associated with the larger patches. Krummel et al. (1987) found that forest patches showed a distinct change in fractal dimension. The reason appears to be that small patches were woodlots whose boundary was affected by human management. the large patches were more complex because they tended to follow natural boundaries, such as topography.
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Albedo reflected sunlight incident sunlight
Initial ponding Pond developing Pond freeze-up Pond peaked
Geometric features of sea ice are captured by the fractal dimensions D, defined by their perimeter P and area A: P ~ D A The complexity of melt ponds on sea ice increases, first gradually, then rapidly, as smaller ponds coalesce to form larger connected regions By analyzing area-perimeter data from many melt ponds, Hohenegger et al. (2012) found an unexpected separation of scales, where the pond fractal dimension D increases rapidly from 1 to 2 as the area crosses a critical value of approximately 100 m 2
Euclidean and Fractal Geometry Things that are fractal are characterized by two distinctive characteristics: 1. Non-whole Dimensions N M D Self similarity dimension Number of smaller self similar objects generated by the iterative process Magnification factor: number each new division must be multiplied by to yield size of original segment D = log N (number of new pieces) log M (Magnification: factor of finer resolution) How much we zoom in on or magnify each new piece to view it the same size as the original. 2. Generated by iteration
Euclidean and Fractal Geometry D = log N (number of new pieces) log M (Magnification: factor of finer resolution) How much we zoom in on or magnify each new piece to view it the same size as the original. Original object a line Divided into 3 new pieces = N Magnification Factor = 3 How much we have to magnify each piece to get object of original size N M D 3 = 3 1 Dimension =
Euclidean and Fractal Geometry D = log N (number of new pieces) log M (Magnification: factor of finer resolution) How much we zoom in on or magnify each new piece to view it the same size as the original. Original object Divided into 9 new pieces = N a square Magnification Factor = 9 How much we have to magnify each piece to get object of original size N M D 9 = 3 2 Dimension =
Euclidean and Fractal Geometry D = log N (number of new pieces) log M (magnification: factor of finer resolution) Original object How much we zoom in on or magnify each new piece to view it the same size as the original. Divided into 27 new pieces = N a cube Magnification Factor = 27 How much we have to magnify each piece to get object of original size N M D 27 = 3 3 Dimension =
Learning Outcomes 7. Self Similarity Self-similarity is patterns, within patterns, within patterns, so that you see complex detail at all scales of observation, all generated by an iterative process. 8. Fractal Geometry There is no typical or average size of events, or objects; they come nested inside each other, patterns within patterns within patterns, all generated by an iterative process. 9. Non-whole Number Dimensions Unlike Euclidian geometry (plane or solid geometry) most natural objects have non-whole number dimensions, something between, for example, 2 and 3.