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Welcome and Course Overview
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How to Take this Course
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Introduction to Functions
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Iteration
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Fixed Points and Stability
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Summary
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Homework (Optional)
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Homework Solutions (Optional)
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Test
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Introduction
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The Idea of the Derivative
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Differential Equations
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Qualitative Solutions
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Computational Solutions
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Summary
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Newton, Laplace, and Determinism
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Homework (Optional)
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Test
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Introducing the Logistic Equation
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Iterating the Logistic Equation
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The Butterfly Effect
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Modeling Styles
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Lyapunov Exponents (Optional)
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Logistic Equation Programs
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Summary
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Homework (Optional)
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Test
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The Logistic Differential Equation
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Logistic Equation with Harvest
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Bifurcation Diagrams
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Bifurcations
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Hysteresis (Optional)
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Summary
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Homework (Optional)
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Test
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Final State Diagrams
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The Bifurcation Diagram
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Bifurcation Diagram Explorations
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Logistic Equation Programs
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Summary
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Homework (Optional)
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Test
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Introduction
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Universality in Maps
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Universality in Physical Systems
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Renormalization
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Summary
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Bifurcation Diagram Programs
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Homework (Optional)
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Further Reading (Optional)
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Test
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Lotka Volterra Differential Equations
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The Phase Plane
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The Hénon Map
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The Lorenz Equations
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Summary
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Hénon Map Program
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Homework (Optional)
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Test
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Further Reading (Optional)
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The Hénon Attractor
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The Lorenz Attractor
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Stretching and Folding
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Links to Programs
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Summary
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Homework (Optional)
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Test
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Diffusion
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Reaction-Diffusion Equations
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Computer Experiments
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Real Experiments
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Summary
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Links to Programs
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Additional Resources (Optional)
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No Homework for Unit 9
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No Test for Unit 9
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Summary and Overview
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Interview: Stephen H. Kellert
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Interview: Stephen W. Morris
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Themes and Conclusions
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Farewell
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6.8 Further Reading (Optional) » References and Additional Reading
Early Papers on Universality and Renormalization:
- Coullet, P., Tresser, C. Itération déndomorphismes et groupe de renormalisation, J. Phys. Colloque C5, C5-25 - C5-28 (1978). pdf
- Feigenbaum, M.J. Quantitative universality for a class of non-linear transformations, J. Stat. Phys. 19 25-52 (1978). pdf
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Cvitanovic, Predrag. Universality in chaos. (2nd edition) Taylor & Francis, 1984. This is a collection of early articles on universality in chaos, the renormalization group, and experimental implications of universality.
Non-technical:
- Gleick, James. Chaos: Making a new science. Random House, 1997. The chapter titled "Universality" is an engaging account of Feigenbaum's discovery of universality. Doesn't say much about what universality and the renormalization group actually are, however.
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Stewart, Ian. Does God play dice?: The new mathematics of chaos. Penguin UK, 1997. Chapter 10 has a very nice, not-very-technical discussion of universality and renormalization.
Technical:
- Peitgen, Heinz-Otto, Hartmut Jürgens, and Dietmar Saupe. Chaos and fractals: new frontiers of science. Springer, 2004. Chapter 11 contains a very clear account of universality and renormalization. Highly recommended.
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Strogatz, Steven. Nonlinear dynamics and chaos: with applications to physics, biology, chemistry and engineering. Westview Press. (2001). Chapter 10 has a nice treatment of the logistic map, including a clear discussion of renormalization.
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Tresser, Charles, Pierre Coullet, and Edson de Faria. Period Doubling. Scholarpedia. September 2013.