Content: 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.
Some questions we will address in this course include:
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 culminate in the famous Picard-Lindelof theorem on existence and uniqueness of solutions of ODEs.
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 behaviour of solutions near critical points - often exactly the regions one is interested in. Different trajectories will be classified and we will develop techniques to answer important questions on the stability properties (or lack thereof) of given solutions.
We will eventually apply these powerful methods to particular examples of practical importance, including the Lotka-Volterra model for the competition between two species and to the Van der Pol and Lienard systems of electrical circuits.
The course will end with a discussion of the Sturm-Liouville theory for solving boundary value problems.
Aims: To extend the knowledge of first year ODEs with a mixture of applications, modelling and theory to prepare for more advanced modules later on in the course.
Objectives: By the end of the course the student should be able to:
- Determine the fundamental properties of solutions to certain classes of ODEs, such as existence and uniqueness of solutions
- Sketch the phase portrait of 2-dimensional systems of ODEs and classify critical points and trajectories
- Classify various types of orbits and possible behaviour of general non-linear ODEs
- Understand the behaviour of solutions near a critical point and how to apply linearization techniques to a non-linear problem
- Apply these methods to certain physical or biological systems
(Complete Lecture Notes will be made available)
Ordinary Differential Equations and Dynamical Systems, Gerald Teschl, [Available online]
Elementary Differential Equations and Boundary Value Problems, Boyce DiPrima 1997
Differential Equations, Dynamical Systems, and an Introduction to Chaos, Hirsch, Smale 2003
Nonlinear Systems, Drazin 1992