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Dynamics, Chaos, and Prediction. Aristotle, 384 – 322 BC. Nicolaus Copernicus, 1473 – 1543. Galileo Galilei , 1564 – 1642. Johannes Kepler , 1571 – 1630. Isaac Newton, 1643 – 1727. Pierre- Simon Laplace, 1749 – 1827. Henri Poincaré , 1854 – 1912. Werner Heisenberg, 1901 – 1976.

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## Dynamics, Chaos, and Prediction

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**Dynamical Systems Theory:**• The general study of how systems change over time • Calculus • Differential equations • Discrete maps • Algebraic topology • Vocabulary of change • The dynamics of a system: the manner in which the system changes • Dynamical systems theory gives us a vocabulary and set of tools for describing dynamics • Chaos: • One particular type of dynamics of a system • Defined as “sensitive dependence on initial conditions” • Poincaré: Many-body problem in the solar system Isaac Newton 1643 – 1727 Henri Poincaré 1854 – 1912**Dr. Ian Malcolm**“You've never heard of Chaos theory? Non-linear equations? Strange attractors?”**Dr. Ian Malcolm**“You've never heard of Chaos theory? Non-linear equations? Strange attractors?”**Chaos in Nature**• Heart activity (EKG) • Computer networks • Population growth and dynamics • Financial data • Dripping faucets • Electrical circuits • Solar system orbits • Weather and climate (the “butterfly effect”) • Brain activity (EEG)**What is the difference between chaos and randomness?**Notion of “deterministic chaos”**A simple example of deterministic chaos:Exponential versus**logistic models for population growth Exponential model: Each year each pair of parents mates, creates four offspring, and then parents die.**Linear Behavior:**The whole is the sum of the parts**Linear Behavior:**The whole is the sum of the parts Linear: No interaction among the offspring, except pair-wise mating.**Linear Behavior:**The whole is the sum of the parts Linear: No interaction among the offspring, except pair-wise mating. More realistic: Introduce limits to population growth.**Logistic model**• Notions of: • birth rate • death rate • maximum carrying capacityk (upper limit of the population that the habitat will support, due to limited resources)**Logistic model**• Notions of: • birth rate • death rate • maximum carrying capacityk (upper limit of the population that the habitat will support due to limited resources) interactions between offspring make this model nonlinear**Logistic model**• Notions of: • birth rate • death rate • maximum carrying capacityk (upper limit of the population that the habitat will support due to limited resources) interactions between offspring make this model nonlinear**Nonlinear behavior of logistic model**birth rate 2, death rate 0.4, k=32 (keep the same on the two islands)**Nonlinear behavior of logistic model**Nonlinear: The whole is different than the sum of the parts birth rate 2, death rate 0.4, k=32 (keep the same on the two islands)**Logistic map**aaa Lord Robert May b. 1936 Mitchell Feigenbaum b. 1944**LogisticMap.nlogo**1. R = 2 2. R = 2.5 3. R = 2.8 4. R = 3.1 5. R = 3.49 6. R = 3.56 7. R = 4, look at sensitive dependence on initial conditions Notion of period doubling Notion of “attractors”**Period Doubling and Universals in Chaos(Mitchell Feigenbaum)**R1 ≈ 3.0: period 2 R2 ≈ 3.44949 period 4 R3 ≈ 3.54409 period 8 R4 ≈ 3.564407 period 16 R5 ≈ 3.568759 period 32 R∞ ≈ 3.569946 period ∞ (chaos)**Period Doubling and Universals in Chaos(Mitchell Feigenbaum)**R1 ≈ 3.0: period 2 R2 ≈ 3.44949 period 4 R3 ≈ 3.54409 period 8 R4 ≈ 3.564407 period 16 R5 ≈ 3.568759 period 32 R∞ ≈ 3.569946 period ∞ (chaos) A similar “period doubling route” to chaos is seen in any “one-humped (unimodal) map.**Period Doubling and Universals in Chaos(Mitchell Feigenbaum)**R1 ≈ 3.0: period 2 R2 ≈ 3.44949 period 4 R3 ≈ 3.54409 period 8 R4 ≈ 3.564407 period 16 R5 ≈ 3.568759 period 32 R∞ ≈ 3.569946 period ∞ (chaos) Rate at which distance between bifurcations is shrinking:**Period Doubling and Universals in Chaos(Mitchell Feigenbaum)**R1 ≈ 3.0: period 2 R2 ≈ 3.44949 period 4 R3 ≈ 3.54409 period 8 R4 ≈ 3.564407 period 16 R5 ≈ 3.568759 period 32 R∞ ≈ 3.569946 period ∞ (chaos) Rate at which distance between bifurcations is shrinking:**Period Doubling and Universals in Chaos(Mitchell Feigenbaum)**In other words, each new bifurcation appears about 4.6692016 times faster than the previous one. R1 ≈ 3.0: period 2 R2 ≈ 3.44949 period 4 R3 ≈ 3.54409 period 8 R4 ≈ 3.564407 period 16 R5 ≈ 3.568759 period 32 R∞ ≈ 3.569946 period ∞ (chaos) Rate at which distance between bifurcations is shrinking:**Period Doubling and Universals in Chaos(Mitchell Feigenbaum)**In other words, each new bifurcation appears about 4.6692016 times faster than the previous one. R1 ≈ 3.0: period 2 R2 ≈ 3.44949 period 4 R3 ≈ 3.54409 period 8 R4 ≈ 3.564407 period 16 R5 ≈ 3.568759 period 32 R∞ ≈ 3.569946 period ∞ (chaos) Rate at which distance between bifurcations is shrinking: This same rate of 4.6692016 occurs in any unimodal map.**Significance of dynamics and chaos for complex systems**• Apparent random behavior from deterministic rules**Significance of dynamics and chaos for complex systems**• Apparent random behavior from deterministic rules • Complexity from simple rules**Significance of dynamics and chaos for complex systems**• Apparent random behavior from deterministic rules • Complexity from simple rules • Vocabulary of complex behavior**Significance of dynamics and chaos for complex systems**• Apparent random behavior from deterministic rules • Complexity from simple rules • Vocabulary of complex behavior • Limits to detailed prediction**Significance of dynamics and chaos for complex systems**• Apparent random behavior from deterministic rules • Complexity from simple rules • Vocabulary of complex behavior • Limits to detailed prediction • Universality

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