**What you will learn**

1) Analyze ODE models arising in science and engineering

2) Understand what is a dynamical system, the relation with ODEs, and the geometric interpretation of ODEs

3) Carry out Poincare's program to analyze the qualitative behavior of an ODE - with focus on low dimensional systems:

4) Use Matlab to perform numerical simulations of ODEs

5) Understand what is "chaos" and its implications

2) Understand what is a dynamical system, the relation with ODEs, and the geometric interpretation of ODEs

3) Carry out Poincare's program to analyze the qualitative behavior of an ODE - with focus on low dimensional systems:

- Identify special structures: conservative, dissipative, Hamiltonian, reversible
- Find landmarks which organize long term behavior: fixed points, closed orbits, attractors, bifurcations (for parametric systems)
- Perform a local analysis around these landmarks to understand the local dynamics around them (linear stability analysis, normal forms, local stable and unstable manifolds)
- Attempt to join the local dynamics to understand the global dynamics (Poincare-Bendixson (n=2), basins of attraction, index theory, global bifurcations)

4) Use Matlab to perform numerical simulations of ODEs

5) Understand what is "chaos" and its implications