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 D. Dobson, 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, we shall provide any backgroung materials needed by the class.

We shall follow Liu'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. 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.

Last updated: 7 / 23 / 9