This course provides a rigorous foundation in the theory of Ordinary Differential Equations (ODEs) and their applications in Dynamical systems. Moving beyond the procedural techniques of undergraduate calculus, we focus on the qualitative behavior, stability, and structural properties of solutions. If time permits, we will learn introductory elements of discrete dynamical systems.

Differential equations (Math 200), Applied linear algebra (Math 203), Applied complex variables (Math 210), and Analysis (Math 311, 312 - suggested).

1. Ordinary Differential Equations and Dynamical Systems (Graduate Studies in Mathematics, 140)
by Gerald Teschl, American Mathematical Society (August 30, 2012), Hardcover: ‎ 356 pages, ISBN-10: ‎0821883283, ISBN-13: ‎978-0821883280

2. Differential Equations, Dynamical Systems & An Introduction to Chaos 3rd Edition

3. Ordinary differential equations with applications, Chicone (Supplementary)

I will cover some topics in the following chapters. Specific plans will be adjusted later.

Key words:

1. Basic existence and uniqueness theorem: Picard iteration, Gronwall's inequality (Ch. 2)
2. Systems of linear differential equations: Phase portraits, fundamental matrix (Ch. 3)
3. Differential equations in the complex domain: Power series solutions and Bessel equations (Ch. 4)
4. Boundary value problem: Sturm-Liouville equations (Ch. 5)
5. Elements of Dynamical Systems (Ch. 6)
6. Planar Dynamical Systems: Poincaré-Bendixson theorem. (Ch. 7)
7. Local hyperbolicity behaviour: Hartman-Grobman Theorem (Ch. 9)

Two lectures per week