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 08 - Index Theory and Introduction to Limit Cycles |
Index of a curve (with respect to a given vector field). Properties of the index. Index of a point. Using index theory to rule out closed trajectories. Some strange things: Index theory in biology. Hairy ball theorem. Combing a torus and connection to tokamaks and fusion. Index theory on compact orientable 2-manifolds. Limit cycles. Examples in science and engineering. Definitions, and why linear systems can't have limit cycles.
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