Math 1110 Differential Equations, Fall 2012

Course Information

A printable pdf version of the syllabus is here.

Basic Information

Instructor: Chris Kottke

Office: 303 Kassar-Gould House


Office hours: Mon 3-4pm, Wed 2-3pm, or by appointment.

Text: Differential Equations, Dynamical Systems, and an Introduction to Chaos, by Hirsch, Smale and Devaney. 2nd Ed (ISBN 978-0-12-349703-1), or 3rd Ed (ISBN 978-0-12-382010-5).
The official textbook is the 2nd edition, used copies of which are in stock at the bookstore. The recently released 3rd edition has been updated to include some additional examples, but is otherwise almost the same. Either one will be fine for this class.



Differential equations are among the most powerful mathematical tools for modeling how systems change in time. While these `systems' come from a wide range of fields including physics, chemistry, electrical engineering, biology, climate science, economics and pure mathematics, they may nevertheless be treated on a common footing using tools we will develop in this course.

In spite of this unification however, the resulting theory is by no means simple! We'll see that even very simple systems can exhibit an extraordinarily complex range of behaviors, including (most famously) the phenomenon known as chaos.

Though there are many different approaches to analyzing and solving differential equations, we take a `dynamical systems' point of view and focus on the study of qualitative behaviors of systems such as stability, limit sets, bifurcations and chaos rather than explicit solution techniques (which tend not to be applicable outside of very specialized situations). The treatment will be rigorous and proofs will be given.

The only formal prerequisite is differential calculus, which we will rely on quite heavily. As we go along we will also develop a considerable amount of linear algebra; though no prior exposure to this is required, it may be beneficial.

Approximate lecture schedule:

1-3Intro to ODE, first order and linear vs. non-linear equations, logistic equation with harvesting.
4-6Systems of equations, reduction of order, planar systems.
7-112D linear algebra, eigenvalues and eigenvectors, classification of planar linear systems.
12-17Higher dimensional linear algebra and systems, matrix exponential. Midterm 1.
18-20Nonlinear dynamical systems, existence/uniqueness, equilibria and linearization.
21-23Bifurcations, nullclines, systems of special type (gradient and Hamiltonian systems).
24-26Limit sets and the Poincare map.
27-32Applications from biology, mechanics, electrical engineering, etc. Midterm 2.
33-36The Lorentz system, intro. to chaos and discrete dynamical systems.

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