MAE5790-1 Course introduction and overview
1:16:31MAE5790-2 One dimensional Systems
1:16:44MAE5790-3 Overdamped bead on a rotating hoop
1:13:12MAE5790-4 Model of an insect outbreak
1:15:16MAE5790-5 Two dimensional linear systems
1:15:20MAE5790-6 Two dimensional nonlinear systems fixed points
1:07:17MAE5790-7 Conservative Systems
1:17:13MAE5790-8 Index theory and introduction to limit cycles
1:13:55Testing for Closed Orbits
1:16:52Van der Pol Oscillator
1:05:58Averaging Theory for Weakly Nonlinear Oscillators
1:16:08Bifurcations in Two-Dimensional Systems
46:54Hopf Bifurcations in Aeroelastic Instabilities and Chemical Oscillators
1:07:57Global Bifurcations of Cycles
1:16:23Chaotic Waterwheel
1:14:12Waterwheel Equations and Lorenz Equations
1:12:42Chaos in the Lorenz Equations
1:16:36Strange Attractor for the Lorenz Equations
1:13:48One-Dimensional Maps
1:14:35Universal Aspects of Period Doubling
1:11:56Feigenbaum's Renormalization Analysis of Period Doubling
1:15:59Renormalization: Function Space and a Hands-On Calculation
1:08:32Universal Aspects of Period Doubling
This lecture investigates the universal aspects of period doubling, using online applets to explore the logistic map's self-similar fractal structure and periodic windows. It introduces Feigenbaum's work, the tangent bifurcation, and universal scaling constants alpha and delta.
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.
One-Dimensional Maps
Next · Lecture 21Feigenbaum's Renormalization Analysis of Period Doubling