Many fundamental problems in the applied sciences reduce to understanding solutions of ordinary differential equations (ODEs). Examples include the laws of Newtonian mechanics, predator-prey models in Biology, and non-linear oscillations in electrical circuits, to name only a few. These equations are often too complicated to solve exactly, so one tries to understand qualitative features of solutions.
When do solutions of ODEs exist and when are they unique? What is the long time behaviour of solutions and can they "blow-up" in finite time? These questions are answered by the Picard Theorem on existence and uniqueness of solutions of ODEs, and its consequences.
The main part of the course will focus on phase space methods. This is a beautiful geometrical approach which often enables one to understand the qualitative behaviour of solutions even when we cannot solve the equations exactly. We will develop techniques to answer important questions about the stability/attraction properties (or instabilities) of given solutions, often fixed points.
We will eventually apply these powerful methods to particular examples of practical importance, including the Lotka-Volterra model for the competition between two species, Hamiltonian systems, and the Lorenz equations, and give an informal introduction to some more advanced topics (e.g. bifurcation theory, Lyapunov exponents).