# MA254 Theory of ODEs

**Lecturer: **James Robinson

**Term(s): **Term 2

**Status for Mathematics students: **List A

**Commitment: **30 one hour lectures

**Assessment: **100% 2 hour examination

**Formal registration prerequisites: **None

**Assumed knowledge:**

- MA131 Analysis or MA137 Mathematical Analysis
- MA133 Differential Equations or MA113 Differential Equations A
- MA106 Linear Algebra - Eigenvectors and eigenvalues of real matrices
- MA134 Geometry & Motion - Parametrisation of curves and geometry of level sets
- MA251 Algebra I - Jordan Canonical Form (this will also be reviewed briefly in the two-dimensional case)

**Useful background:** Some ideas introduced in MA259 Multivariable Calculus (open and closed sets, compactness) and expanded on in MA260 Norms, Metrics, & Topologies or MA222 Metric Spaces will be used throughout, although a review of the main concepts will be available. ["Assumed knowledge of which you will be reminded."] The module will make use of other parts of MA260 Norms, Metrics, & Topologies or MA222 Metric Spaces too, but as this is running concurrently the required concepts will be introduced during the module.

**Synergies: **

- MA261 Differential Equations: Modelling and Numerics
- MA250 Introduction to PDEs
- MA209 Variational Principles

**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