procedural methods • model chaos nothpcg.purdue.edu/.../lectures/cs590-cgs-06-chaos.pdf ·...
TRANSCRIPT
© Bedrich Benes
CS 590 – CGSProcedural MethodsChaos
Bedrich Benes, Ph.D.Purdue UniversityDepartment of Computer Science
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Procedural Techniques• Model is generated by a piece of code.
• Model is not represented as data!
• The generation can take some time
• and of course the data can be pre‐calculated
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Procedural TechniquesThree huge (vague) classes:• Fractals• Particle systems• GrammarsUsed when shape cannot be represented as a surface (fire, water, smoke, flock of birds, explosions, model of mountain, grass, clouds, plants, facades, cities, etc.)
Used for Simulation of Natural Phenomena
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The Logistic MapMay, R. M. (1976). Simple mathematical models with very complicated dynamics. Nature, 261(5560), 459‐467.
• The logistic map• Verhulst dynamics (Pierre Verhulst, 18041849)• It models the population growth• Having a population of 𝑥
with a growth rate 𝑟 (biotic potential)what will be the population at 𝑥 ?
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The Logistic Map
• Would lead to exponential growth – no limit• But, if the population consumes its resources, it will also die off.
• Assume is normalized • Let’s multiply by normalized • The logistic map is:
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The Logistic Equation• The logistic equation assumes continuous time• The equation has form
• The logistic map is its iterated version for
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The Logistic Map • It is an upside down parabola
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The Logistic Map • How does and ?
r 0.00 0.30 0.40 0.60 0.70 0.90 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60 2.80 3.00 3.10 3.20 3.30x0 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10x1 0.00 0.03 0.04 0.05 0.06 0.08 0.09 0.11 0.13 0.14 0.16 0.18 0.20 0.22 0.23 0.25 0.27 0.28 0.29 0.30x2 0.00 0.01 0.01 0.03 0.04 0.07 0.08 0.12 0.15 0.20 0.24 0.30 0.35 0.41 0.47 0.53 0.59 0.62 0.66 0.69x990 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.17 0.29 0.38 0.44 0.50 0.55 0.58 0.62 0.64 0.66 0.56 0.51 0.48x991 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.17 0.29 0.38 0.44 0.50 0.55 0.58 0.62 0.64 0.67 0.76 0.80 0.82x992 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.17 0.29 0.38 0.44 0.50 0.55 0.58 0.62 0.64 0.66 0.56 0.51 0.48x993 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.17 0.29 0.38 0.44 0.50 0.55 0.58 0.62 0.64 0.67 0.76 0.80 0.82x994 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.17 0.29 0.38 0.44 0.50 0.55 0.58 0.62 0.64 0.66 0.56 0.51 0.48x995 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.17 0.29 0.38 0.44 0.50 0.55 0.58 0.62 0.64 0.67 0.76 0.80 0.82x996 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.17 0.29 0.38 0.44 0.50 0.55 0.58 0.62 0.64 0.66 0.56 0.51 0.48x997 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.17 0.29 0.38 0.44 0.50 0.55 0.58 0.62 0.64 0.67 0.76 0.80 0.82x998 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.17 0.29 0.38 0.44 0.50 0.55 0.58 0.62 0.64 0.66 0.56 0.51 0.48x999 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.17 0.29 0.38 0.44 0.50 0.55 0.58 0.62 0.64 0.67 0.76 0.80 0.82
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The Logistic Map
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The Logistic Map • How does and ?
5• Cobweb plot
Boeing, G. Visual Analysis of Nonlinear Dynamical Systems: Chaos, Fractals, Self-Similarity and the Limits of Prediction. Systems 2016, 4, 37.
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The Logistic Map • How does and ?
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• Cobweb plot
Boeing, G. Visual Analysis of Nonlinear Dynamical Systems: Chaos, Fractals, Self-Similarity and the Limits of Prediction. Systems 2016, 4, 37.
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The Logistic Map • How does and ?
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• Cobweb plot• Doubling the results
Boeing, G. Visual Analysis of Nonlinear Dynamical Systems: Chaos, Fractals, Self-Similarity and the Limits of Prediction. Systems 2016, 4, 37.
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The Logistic Map 0 𝑟 1 𝑥 → 01 𝑟 2 𝑥 → (quick approach)2 𝑟 3 𝑥 → (fluctuation first)3 𝑟 1 √6 𝑥 → 𝑥 or 𝑥 (bifurcation)3.44949 𝑟 3.54409 four values oscillationincreasing 𝑟 oscillation 8, 16, 32,…The length of the interval 𝑟 /𝑟 𝛿 for period doubling cascade
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The Logistic Map • How does and ?
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• Deterministic chaos
Boeing, G. Visual Analysis of Nonlinear Dynamical Systems: Chaos, Fractals, Self-Similarity and the Limits of Prediction. Systems 2016, 4, 37.
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The Logistic Map • The algorithm:1) Run the model for random
2) For each run, forget the first 500 numbers
3) Draw the plot
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Bifurcation Diagram
By Jordan Pierce ‐ Own work, CC0, https://commons.wikimedia.org/w/index.php?curid=1644522916
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Bifurcation Diagram
By InXnI - Own work, CC0, https://commons.wikimedia.org/w/index.php?curid=18744163 17© Bedrich Benes
Bifurcation Diagram
By Biajojo - Own work, CC0, https://commons.wikimedia.org/w/index.php?curid=33573458 18
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The Logistic Map • For the bifurcation can reach any value• Period‐doubling to chaos• 8‐16‐32• For it seems to jump randomly to any value• The 2D structure is self‐similar
it repeats patterns at zoom levels
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The Logistic Map Feigenbaum constant 𝛿 4.669 20160910299067185320382157866920Expresses the ratio of bifurcation of a non‐linear map𝛿 lim→ 𝐿𝐿 𝑎 𝑎𝑎 𝑎𝐿 𝑎 ,𝑎 is the period𝛿 represents the period doublingIt is believed to be transcendental
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The Logistic Map Feigenbaum constant 𝛿 4.66920Mitchel Jay Feigenbaum (1944‐2019)
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Poincare Map (State space)axis is axis is
Boeing, G. Visual Analysis of Nonlinear Dynamical Systems: Chaos, Fractals, Self-Similarity and the Limits of Prediction. Systems 2016, 4, 37.
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Poincare Mapaxis is axis is
Boeing, G. Visual Analysis of Nonlinear Dynamical Systems: Chaos, Fractals, Self-Similarity and the Limits of Prediction. Systems 2016, 4, 37.
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Butterfly Effect• Sensitivity to initial conditions • If the initial conditions change a bit
the process can diverge significantly• First observed in weather simulation by
Edward Lorenz (1917‐2019)
“the present determines the future, but the approximate present does not approximately determine
the future”
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Butterfly Effect
Boeing, G. Visual Analysis of Nonlinear Dynamical Systems: Chaos, FractalsSelf-Similarity and the Limits of Prediction. Systems 2016, 4, 37. 25