Lecturer: Ian Melbourne

Term(s): Term 2

Status for Mathematics students: List A (MA254 is for 2nd years, MA354 for 3rd years provided MA254 has not been taken in a previous year).

Commitment: 30 one hour lectures

Assessment: 100% 2 hour examination

Formal registration prerequisites: None

Assumed knowledge:

• MA142 Calculus 1
• MA133 Differential Equations
• MA148 Vectors and Matrices - Eigenvectors and eigenvalues of real matrices
• MA133 Differential Equations - Parametrisation of curves and geometry of level sets

Useful background: Some ideas about open and closed sets, compactness introduced in previous analysis modules and expanded on in MA222 Metric Spaces will be used throughout. Since this is running concurrently, the required concepts will be introduced during the module.

Synergies:

• MA222 Metric Spaces
• MA2K4 Numerical Methods and Computing
• MA250 Introduction to PDEs

Leads to: The following modules have this module listed as assumed knowledge or useful background:

• MA3J4 Mathematical Modelling with PDE
• MA3H5 Manifolds
• MA3H7 Control Theory
• MA390 Topics in Mathematical Biology
• MA3J3 Bifurcations, Catastrophes and Symmetry
• MA4J1 Continuum Mechanics
• MA4C0 Differential Geometry
• MA4E0 Lie Groups
• MA4H0 Applied Dynamical Systems
• MA4M1 Epidemiology by Example

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

Books:
(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

Additional Resources