MATH 6307 Ordinary Differential Equations 1
Fall 2020
TR 14:00-15:15, Remote synchronous
Office Hours: TR 13:00-14:00, Remote synchronous.
General Information
Description
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.
Prerequisites:
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
Textbook
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
Dover
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:
- General Properties Existence, uniqueness, continuous dependence, Liapunov functions, attractors, chain recurrence, Morse decomposition, Poincaré-Bendixson theorem
- Linear Systems Stability and perturbations, periodic systems, nonhomogeneous systems, Fredholm alternative, Hamiltonian systems, mappings
- Local Theory of Equilibria Hartman-Grobman theorem, stable and unstable manifolds, foliations, center manifolds, elementary bifurcations
- Bifurcations Poincare-Andronov-Hopf bifurcation, behavior near a homoclinic
Class development:
First week
- 1.1 Definition and Notation.
- 1.2 Existence and Uniqueness. (Notes: 1.1, 1.2 and 1.3)
- 1.3 Gronwall's Inequalities. (Notes: 1.4)
- 2.1 Phase Space, Orbits.
Second week
- 2.2 Critical Points and Linearization.
- 2.4 First Integral and Integral of Motion
- 2.5 Evolution of a Volume Element, Liouville Theorem
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
- Stable and Unstable Manifolds: proof of the theorem. The codes used in class are available on Canvas.
- Appendix A
- 4.1 Bendixon's criterion.
Fifth week
- 4.2 Geometric Auxiliaries.
- 4.3 The Poincare-Bendixon theorem.
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
- 4.4 Application of the Poincare-Bendixon theorem.
- 4.5 Periodic solution in Rn
- 5.1 Simple Example.
Seventh Week.
- 5.2 Stability of Equilibrium Solution.
- 5.3 Stability of Periodic Solution.
- 5.4 Linearization.
Eighth Week.
- 6.1 Equation with Constant Coefficients.
- 6.2 Equation with Coefficients that have a Limit.
- 6.3 Equation with Periodic Coefficients.
Nineth Week.
- 7.1 Asymptotic stability of the trivial solution.
- 7.2 Instability of the trivial solution.
- 7.3 Stability of periodic solutions of autonomous systems.
Tenth Week.
- 8.1 Introduction.
- 8.2 Lyapunov functions.
- 8.3 Hamiltonian system and system with first integrals.
Third Homework: from the textbook ex: 6.3, 6.7, 7.2, 7.7. Homework due on November 2.
Ideas for Projects
- Elliptic Orbits in Billiards: Article by V. Donnay
- Variational Principle in Mechanics: Chapter 2.24
- Spin-orbit Resonances: selection by dissipation: Draft by me, Gallavotti and Gentile
- Anchor Escapement Mechanism:Chapter 217-2.19
- Epidemiological models: Theory and Applications
- Immunological models: General Theory and Examples
- Differential Equation with Discontinuities: General Theory and Examples