MATH 6307 Ordinary Differential Equations 1

Fall 2020

TR 14:00-15:15, Remote synchronous

Office Hours: TR 13:00-14:00, Remote synchronous.

Professor Federico Bonetto

General Information


This is the first classes in a series of two. The sequence develops the qualitative theory for systems of ordinary differential equations. Topics will include stability, Lyapunov functions, Floquet theory, attractors, invariant manifolds, bifurcation theory, normal forms.


MATH 4541 and MATH 4542.

Course Goals and Learning Outcomes;

My goal is to teach you how to recognize a problem than can be described by a system of differential equation. Once the proper system of equation is written you will learn how to see if it is well posed (local/global existence and uniqueness). We will review the properties solution of linear systems as a basic block for more complex system. You will then learn how to analyze qualitatively the behavior of solutions starting from fixed point and periodic orbits, their stability and bifurcation and finally their invariant manifolds. In the second semester we will move to perturbation theory and chaos.

Course Modality:

The course will be remote synchronous. Class will be held using Bluejeans (my favourite) or Teams. We will experiment with this at the beginning of class.

Course Requirements and Grading

There will be two midterm and one final. The midterms and final will be in the take home format. I will post a set of exercises meant to use the material learnt in class to analyze a concrete problem. You will have several days to work out the exercises. During this period I'll be available to help and direct your work. Hopefully, in this way, these assignments will be an opportunity to learn. The date of the midterms will be announced shortly.
I will also assign homework from the textbook. This are mostly meant to check that you are following along with the class material.
The final grade will be based on HWs (10%), midterms (50%) and final (40%).
We will also discuss personal projects to be completed together with the final. To prepare a good project it is necessary to start as soon as possible. I can propose subjects for possible projects but I'd prefer if you find a problem involving differential equation you are interested in and that can be analyzed with the tools you will learn in class.

Course Materials


The class text book is:

Nonlinear Differential Equation and Dynamical System
Ferdinand Verhulst
Springer, 2nd edition

I'll also use the lecture notes by prof. Jack Hale. The notes will posted on canvas and the class webpage.
This notes are an update and extension of the book:

Ordinary Differential Equations.
Jack K. Hale

It will not be necessary to buy Hale's book, but I strongly suggest to get it also because is a great book.

Web page

The weekly evolution of the class will be posted on the class web page, together with HW and extra class material. You can also check the we pages of previous edition of this class: Fall 2015 and Fall 2016

Topic Outline:

Class development:

First week

Second week

First Homework: from the textbook ex: 2.2, 2.4, 2.6, 3.2, 3.5. Homework due on September 10.

Third week

Fourth week

Fifth week

Second Homework: from the textbook ex: 4.2, 4.5, 4.6, 4.8. Homework due on September 27.

First midterm will be posted on Monday October 5.

Sixth Week

Seventh Week.

Eighth Week.

Nineth Week.

Tenth Week.

Third Homework: from the textbook ex: 6.3, 6.7, 7.2, 7.7. Homework due on November 2.

Ideas for Projects

  1. Elliptic Orbits in Billiards: Article by V. Donnay
  2. Variational Principle in Mechanics: Chapter 2.24
  3. Spin-orbit Resonances: selection by dissipation: Draft by me, Gallavotti and Gentile
  4. Anchor Escapement Mechanism:Chapter 217-2.19
  5. Epidemiological models: Theory and Applications
  6. Immunological models: General Theory and Examples
  7. Differential Equation with Discontinuities: General Theory and Examples