I will also assign homework from the textbook. This are mostly meant to check that you are following along with the class material.

The final grade will be based on HWs (10%), midterms (50%) and final (40%).

We will also discuss personal projects to be completed together with the final. To prepare a good project it is necessary to start as soon as possible. I can propose subjects for possible projects but I'd prefer if you find a problem involving differential equation you are interested in and that can be analyzed with the tools you will learn in class.

Nonlinear Differential Equation and Dynamical System

Ferdinand Verhulst

Springer, 2nd edition

I'll also use the lecture notes by prof. Jack Hale. The notes will posted on canvas and the class webpage.

This notes are an update and extension of the book:

Ordinary Differential Equations.

Jack K. Hale

Dover

It will not be necessary to buy Hale's book, but I strongly suggest to get it also because is a great book.

- General Properties Existence, uniqueness, continuous dependence, Liapunov functions, attractors, chain recurrence, Morse decomposition, PoincarĂ©-Bendixson theorem
- Linear Systems Stability and perturbations, periodic systems, nonhomogeneous systems, Fredholm alternative, Hamiltonian systems, mappings
- Local Theory of Equilibria Hartman-Grobman theorem, stable and unstable manifolds, foliations, center manifolds, elementary bifurcations
- Bifurcations Poincare-Andronov-Hopf bifurcation, behavior near a homoclinic

**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 (extra material on Contraction Map, general consideration on the Stable Manifold Theorem, proof of the Stable Manifold Theorem).

**Fourth week**

- Stable and Unstable Manifolds: proof of the theorem. The codes used in class are available on Canvas.
- Appendix A
- 4.1 Bendixon's criterion.

**Fifth week**

- 4.2 Geometric Auxiliaries.
- 4.3 The Poincare-Bendixon theorem.

**First midterm will be posted on Monday October 5.**

**Sixth Week**

- 4.4 Application of the Poincare-Bendixon theorem.
- 4.5 Periodic solution in Rn
- 5.1 Simple Example.

**Seventh Week.**

- 5.2 Stability of Equilibrium Solution.
- 5.3 Stability of Periodic Solution.
- 5.4 Linearization.

**Eighth Week.**

- 6.1 Equation with Constant Coefficients.
- 6.2 Equation with Coefficients that have a Limit.
- 6.3 Equation with Periodic Coefficients.

**Nineth Week.**

- 7.1 Asymptotic stability of the trivial solution.
- 7.2 Instability of the trivial solution.
- 7.3 Stability of periodic solutions of autonomous systems.

**Tenth Week.**

- 8.1 Introduction.
- 8.2 Lyapunov functions.
- 8.3 Hamiltonian system and system with first integrals.

**Ideas for Projects**

- Elliptic Orbits in Billiards: Article by V. Donnay
- Variational Principle in Mechanics: Chapter 2.24
- Spin-orbit Resonances: selection by dissipation: Draft by me, Gallavotti and Gentile
- Anchor Escapement Mechanism:Chapter 217-2.19
- Epidemiological models: Theory and Applications
- Immunological models: General Theory and Examples
- Differential Equation with Discontinuities: General Theory and Examples