MAE 5790: Nonlinear Dynamics and Chaos
MAE 5790: Nonlinear Dynamics and Chaos (Spring 2014, Cornell University). Instructor: Professor Steven Strogatz. This course provides an introduction to nonlinear dynamics, with applications to physics, engineering, biology, and chemistry. It closely follows Prof. Strogatz's book, "Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering."
The mathematical treatment is friendly and informal, but still careful. Analytical methods, concrete examples, and geometric intuition are stressed. The theory is developed systematically, starting with first-order differential equations and their bifurcations, followed by phase plane analysis, limit cycles and their bifurcations, and culminating with the Lorenz equations, chaos, iterated maps, period doubling, renormalization, fractals, and strange attractors.
| Lecture 20 - Universal Aspects of Period Doubling |
Exploring the logistic map and period doubling with online applets. Interactive cobweb diagrams. Interactive orbit diagram. Zooming in to see the periodic windows. Self-similar fractal structure: each periodic window contains miniature copies of the whole orbit diagram. Smooth curves running the orbit diagram: supertracks. How are periodic windows born? Example: Birth of period three. Tangent bifurcation. The mechanism underlying the fractal structure.
Introduction to Mitchell Feigenbaum's work on universality, what he found, and why it matters. Testable predictions about periodic doubling in physical and chemical systems. Sine map vs. logistic map. The universal scaling constants alpha and delta.
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