### Textbook

The class text book is:

Nonlinear Differential Equation and Dynamical System
Ferdinand Verhulst
Springer, 2nd edition

I'll also use the lecture notes by prof. Jack Hale. The notes are linked below.
The notes are an update and extension of the book:

Ordinary Differential Equations.
Jack K. Hale
Dover

### Syllabus

See the online syllabus.

There will be one midterm and one final. The date of the midterm will be announced shortly. The final will be, at least in part, project based. I will assign HW and but I will not collect them. The final grade will be based on midterms (40%) and final (60%).

First week

• 1.1 Definition and Notation.
• 1.2 Existence and Uniqueness. (Notes: 1.1, 1.2 and 1.3)
• 1.3 Gronwall's Inequalities. (Notes: 1.4)
• 2.1 Phase Space, Orbits.

Second week

• 2.2 Critical Points and Linearization.
• 2.4 First Integral and Integral of Motion
• 2.5 Evolution of a Volume Element, Liouville Theorem

Third week

• 3.1 Two Dimensional Linear System
• 3.2 Remark on Three-dimensional linea systems
• 3.3 Critical Point of Nonlinear System

First Homework: from the textbook ex: 2.2, 2.4, 2.6, 3.2, 3.5. Homework are optional. You can turn them in and I'll grade them.

Fourth week

• Stable and Unstable Manifolds: note coming soon.
• Appendix A

Fifth week

• 4.1 Bendixon's criterion.
• 4.2 4.2 Geometric Auxilliaries.

First midterm will be on Monday October 17.

Sixth-Seventh Week.

• 4.3 The Poincare-Bendixon theorem.
• 4.4 Application of the Poincare-Bendixon theorem.
• 4.5 Periodic solution in Rn
Second Homework: from the textbook ex: 4.2, 4.5, 4.6, 4.8.

Last year midterm with solution set.

Eighth Week.
• 5.1 Simple Example.
• 5.2 Stability of Equilibrium Solution.
• 5.3 Stability of Periodic Solution.
• 5.4 Linearization.
Nineth Week.
• 6.1 Equation with Constant Coefficients.
• 6.2 Equation with Coefficients that have a Limit.
• 6.3 Equation with Periodic Coefficients.
Solution set for the first midterm.

Ideas for Projects
1. Elliptic orbits in Billiards: Article by V. Donnay
2. Variational principle in Mechanics: Chapter 2.24
3. Spin-orbit resonances: selection by dissipation: Draft by me, Gallavotti and Gentiule
4. Anchor Escapement mechanism:Chapter 217-2.19
5. More to come

Final exam due on December 16 by midnight. Please read it asap and let me know about any issue with the questions.
There are 8 questions worth from 20 to 30 points each. A total of 100 points obtained through completely solved questions will be considered a perfect score.
You should return it, possibly via email, by Friday 16 midnight. Any text arriving later will not be considered.
If you decided to develop a personal project, you can make an appointment to present it on Monday Tuesday or Wednesday. If you decide not to present it, you can email a report to me by Friday 16 midnight. Any report arriving later will not be considered.
Yuou should return (or present) at least one between a project and the take home final.