Course Title: Ordinary Differential Equations MATH 6410 - 1 Andrejs Treibergs http://www.math.utah.edu/~treiberg/M6414.html M, W, F at 2:00 - 2:50 in JWB 308 10:45 - 11:35 M, W, F, in JWB 224 (tent.) and by appointment treiberg@math.utah.edu Math 5210, its equivalent or consent of instructor. Lawrence Perko, Differential Equations and Dynamical Systems, Springer Texts in Applied Mathematics 7, 1991. 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