## Math 519: Ordinary Differential Equations

### Instructor

Prof. Gloria Mari Beffa; Office: Van Vleck 309; e-mail: maribeff@math.wisc.edu

### Office hours

Monday 10-11; Wednesday 1-2

### Course

Math 519, meets Monday-Wednesday-Friday 8:50-9:40am in B 219.

### Prerequisite:

Math 319 or 320 or 340, & Math 421 or 521; or cons inst.

### Textbook:

Differential Equations, Dynamical Systems, and an Introduction to Chaos by Morris Hirsch and Stephen Smale and Robert Devaney. Third Edition. Academic Press, 2013.

### Description:

The course will be a rigorous self contained introduction to ordinary differential equations intended for undergraduate math majors and advanced or graduate students from economics, engineering, physics. Topics will include theory of linear systems based on linear algebra, proof of basic existence theorems, stability theory, bifurcations and applications to mechanical and biological systems.

### Rough outline of content:

• 1. Introduction (chapter 1). Review of Linear algebra (chapter 5).
• 2. Complex and real vector spaces. Solution of linear systems with complex distinct eigenvalues. Real normal form of a matrix (chapter 5).
• 3. Multiple eigenvalues. Generalized eigenspaces. Complex and real Jordan normal form (chapter 5).
• 4. Review of topology in \$R^n\$. Exponentials of matrices (chapter 6).
• 5. Stability for linear systems. Topological classification of hyperbolic systems (chapter 6).
• 6. Linear non-autonomous systems. Fundamental matrix. Periodic systems and Floquet theory (chapter 6).
• 7. Review of uniform convergence and contraction principle. Proof of the local existence theorem (chapter 7).
• 8. Continuity of solutions in initial conditions. Differentiability in initial conditions (chapter 7).
• 9. Maximal solutions of differential equations. The flow of the differential equation. Flow box theorem (chapter 7/10).
• 10. Stability of equilibria. Lyapunov functions. Stability by linearization (chapter 8 and 9).
• 11. Hyperbolic equilibria. Grobman-Hartman theorem (without proof). Stable and unstable invariant manifolds.
• 12. Periodic orbits. Poincare map. Stability of periodic orbitsi (chapter 10).
• 13. Limit sets and attractors. Poincare-Bendixson theory (chapter 10).
• 14. Bifurcations for equilibria and periodic orbits. Period doubling and Andronov-Hopf bifurcation (chapter 10).
• 15. Applications to Classical Mechanics (if time allows).
• 16. Applications to Biological Systems (if time allows).