Math 6410 § 1 - - - Home Page - -

Course Title:	Ordinary Differential Equations
Course Number:	MATH 6410 - 1
Instructor:	Andrejs Treibergs
Home Page:	`http://www.math.utah.edu/~treiberg/M6414.html`
Place & Time:	M, W, F at 2:00 - 2:50 in JWB 308
Office Hours:	10:45 - 11:35 M, W, F, in JWB 224 (tent.) and by appointment
E-mail:	`treiberg@math.utah.edu`
Prerequisites:	Math 5210, its equivalent or consent of instructor.
Main Text:	Lawrence Perko, Differential Equations and Dynamical Systems, Springer Texts in Applied Mathematics 7, 1991.
Supplementary Notes:	Christopher Grant, Theory of Ordinary Differential Equations, pdf, Solutions. List of additional supplementary materials used in the course: `M6414Supplement.html` .

In this first semester of a year long graduate course in differential equations, we shall focus on ordinary differential equations and dynamical systems. The second semester, Math 6420 taught by P. Bressloff, will emphasize partial differential equations. In this course, along with the Math 6420, we shall try to cover the syllabus for the qualifying exam in differential equations. Although some mathematical sophistication is required to take the course, and it moves at the blazing speed of a graduate course, I shall provide any backgroung materials needed by the class.

Outline

We shall follow Perko's text covering the behavior of solutions: existence and uniqueness, continuous dependence on data; and dynamical systems properties: long time existence, stability theory, Floquet theory, invariant manifolds and bifurcation theory. We shall discuss as many applications as we can. Topics include (depending on time):

Introduction to ODE. Applications. Review of calculus.
Linear systems and stability.
Existence, uniqueness and continuity theorems.
Qualitative theory, Lipunov stability, Limit sets and attractors.
Applications to physical and biological systems. Charged particle, coupled pendula, planetary systems.
Invariant manifolds and linearizations. Hartman-Grobman theorem.
Planar flows. Poincaré-Bendixon theory.
Periodic solutions and their stability.
Sturm-Liouville Theory.
Bifurcation Theory.
Chaos.

Expected Learning Outcomes

At the end of the course the student is expected to master the theorems, methods and applications of

Local Theory

Proof of the Local Existence and Uniqueness Theorem for ODE's
Continuation of solutions
Gronwall's Inequality
Dependence of the solution on parameters
Contraction Mapping Principle

Linear Equations

Linear systems with constant coefficients
Jordan Normal Forms
Matrix exponential and logarithm, Fundamental Solution
Variation of Parameters Formula
Floquet Theory for periodic equations

Stability

Liapunov Stability, Assymptotic Stability of solutions
Liapunov Functions
Proof of the Linearized Stability for Rest Points Theorem
Grobman - Hartman Theorem
Stable and Center Manifold Theorems

Dynamical Systems

Omega limit sets and limit cycles
Poincaré-Bendixson Theorem
Poincaré Map and stability of periodic orbits

Bifurcation Theory

Persistence of periodic orbits
Normal forms for saddle-node, transcritical and pitchfork bifurcations
Hopf Bifurcation

Grading

The success of the student will be measured by graded daily homework.

In addition, the student's performance will be reported to the Graduate Committee, which decides the continuation of financial support annually. Ultimately, the learning will also be measured by the Differential Equation Qualifying Examination.

Last updated: 6 - 20 - 12