Linearization. Jacobian matrix. Borderline cases. Example: Centers are delicate. Polar coordinates. Example of phase plane analysis: rabbits versus sheep (Lotka-Volterra model of competition in population biology). Stable manifold of a saddle point.

Reading: Strogatz, "Nonlinear Dynamics and Chaos", Sections 6.3, 6.4.

1. Course Introduction and Overview
2. One Dimensional Systems
3. Overdamped Bead on a Rotating Hoop
4. Model of an Insect Outbreak
5. Two Dimensional Linear Systems
6. Two Dimensional Nonlinear Systems: Fixed Points
7. Conservative Systems
8. Index Theory and Introduction to Limit Cycles
9. Testing for Closed Orbits
10. Van der Pol Oscillator
11. Averaging Theory for Weakly Nonlinear Oscillators
12. Bifurcations in Two Dimensional Systems
13. Hopf Bifurcations in Aeroelastic Instabilities and Chemical Oscillators
14. Global Bifurcations of Cycles
15. Chaotic Waterwheel
16. Waterwheel Equations and Lorenz Equations
17. Chaos in the Lorenz Equations
18. Strange Attractor for the Lorenz Equations
19. One Dimensional Maps
20. Universal Aspects of Period Doubling
21. Feigenbaum's Renormalization Analysis of Period Doubling
22. Renormalization: Function Space and a Hands-on Calculation
23. Fractals and the Geometry of Strange Attractors
24. Hénon Map
25. Using Chaos to Send Secret Messages

This course of 25 lectures, filmed at Cornell University in Spring 2014, is intended for newcomers to nonlinear dynamics and chaos. 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.

A unique feature of the course is its emphasis on applications. These include airplane wing vibrations, biological rhythms, insect outbreaks, chemical oscillators, chaotic waterwheels, and even a technique for using chaos to send secret messages. In each case, the scientific background is explained at an elementary level and closely integrated with the mathematical theory. The theoretical work is enlivened by frequent use of computer graphics, simulations, and videotaped demonstrations of nonlinear phenomena.

The essential prerequisite is single-variable calculus, including curve sketching, Taylor series, and separable differential equations. In a few places, multivariable calculus (partial derivatives, Jacobian matrix, divergence theorem) and linear algebra (eigenvalues and eigenvectors) are used. Fourier analysis is not assumed, and is developed where needed. Introductory physics is used throughout. Other scientific prerequisites would depend on the applications considered, but in all cases, a first course should be adequate preparation.