Non-Linear Differential Equations

Introduction

This course provides an introduction to applied non-linear ordinary differential equations. This course is applied mathematics. There will be no technical lemmas or abstract definitions!

The course typically consists of twelve two-hour lectures. Participants will spend most of their time working on example problems and tutorial sheets, often using computer packages such as maple and matlab. In some years, towards the end of the session, students are given a project to apply the ideas that they have learnt.

A set of AMSI guidelines for this course are available here.

Content

Topics to be covered include (but are not limited to):

First-order differential equation: Graphical insights, steady-state solutions and their stability, steady-state diagrams and bifurcations.
Singularity theory with a distinguished parameter: singularity theory and bifurcation points, constructing static bifurcation diagrams.
Systems of two first-order differential equations
1. steady-state solutions and their stability: local and Liapunov.
2. the absence of periodic solutions: Bendixon's Criteria and Dulac's Test.
3. periodic behaviour: the Hopf bifurcation Theorem, sub-critical and super-critical Hopf bifurcations.
4. bifurcations and steady-state diagrams: singularity theory and bifurcation points.
5. degenerate Hopf bifurcations: the double Hopf bifurcation, the Bautin bifurcation, the double-zero eigenvalue bifurcation.

Prerequisites

No knowledge of applied mathematics is assumed. Little knowledge above second year calculus is required. It will be assumed that you have used Maple previously. If you don't like Maple you are free to use an equivalent package. The main skill that is required is mathematical maturity in knowing how to approach problems.

Assessment

Your final mark in MATH971 will be determined as follows. Two marks will be calculated using scheme one (S1) and scheme two (S2).

Scheme	S1	S2
Final Exam	60	50
Assignments	40	50

Your final mark will be the higher of the marks calculated using schemes one and two. Scaling of marks is not a standard procedure in this subject.

Note that you are not required to `pass' each individual component to receive a pass grade in MATH971. However, you would seriously jeopardise your chances of passing this subject if you do not aim to be successful in every component of the assessment.