Course Descriptions

16:640:515 - Ordinary Differential Equations

Marcio Gameiro

 Course Description:

This is an introduction to the theory of ordinary differential equations. We will cover the classical results: existence and uniqueness theorems; linear theory including Floquet theory and elementary bifurcations; stable and unstable manifolds; boundary value problems; and a brief introduction to chaotic dynamics. The novelty of the course is that the proofs will be presented in a manner which allows for rigorous computer verification. Using Julia and Matlab we will apply these new techniques to rigorously extract specific solutions and explore the dynamics of explicit nonlinear systems. For a more detailed overview of the philosophy of the course (we will only consider ODEs as opposed to PDEs and FDEs) see (http://www.ams.org/notices/201509/rnoti-p1057.pdf) and the description of the AMS short course delivered at the National Meeting January 2015 (http://www.ams.org/notices/201509/rnoti-p1106.pdf) To have a sense of the cutting edge work see: http://crm.math.ca/camp-nonlinear/

Text:

Ordinary Differential Equations: A Constructive Approach by J.-B. van den Berg, M. Gameiro, J.-P. Lessard, J. Mireles-James, and K. Mischaikow.

Prerequisites:

An undergraduate course on ordinary differential equations, linear algebra, advanced calculus and undergraduate analysis.

 

Previous Semesters:


SPRING 2019

Schedule of Sections:

Konstantin Mischaikow

Text:

Ordinary Differential Equations: A Constructive Approach by J.-B. van den Berg, M. Gameiro, J.-P. Lessard, J. Mireles-James, and K. Mischaikow.

Prerequisites:

An undergraduate course on ordinary differential equations, linear algebra, advanced calculus and undergraduate analysis.

Description:

This is an introduction to the theory of ordinary differential equations.

We will cover the classical results: existence and uniqueness theorems; linear theory including Floquet theory and elementary bifurcations; stable and unstable manifolds; boundary value problems; and anintroduction to chaotic dynamics.

This is an analysis course, however the novelty is that the proofs will be presented in a manner which allows for rigorous computer verification.

Using Julia we will apply these new techniques to rigorously extract specific solutions and explore the dynamics of explicit nonlinear systems.

 
 
For a more detailed overview of the philosophy of the course (we will only consider ODEs as opposed to PDEs and FDEs) see
http://www.ams.org/notices/201509/rnoti-p1057.pdf
and the description of the AMS short course delivered at the National Meeting January 2015
http://www.ams.org/notices/201509/rnoti-p1106.pdf
To have a sense of the cutting edge work see
http://www.crm.math.ca/crm50/en/activities/2019-activities/topological-and-rigorous-computational-methods-for-high-dimensional-dynamics/