# MATH971: Applied Non-Linear Differential Equations

This page contains material for the course: MATH 971 Applied Non-Linear Differential Equations.

The lecture notes are those of the 2009 version of the course and should not be downloaded by students taking the course in 2010.

## 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 outside of lectures 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):

1. First-order differential equation: Graphical insights, steady-state solutions and their stability, steady-state diagrams and bifurcations.
2. Singularity theory with a distinguished parameter: singularity theory and bifurcation points, constructing static bifurcation diagrams.
3. 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.

In Chapter One we consider a single autonomous first-order differential equation. Given the ODE dx/dt = f(x) we can understand everything of interest by graphing the function y = f(x). This leads to the concepts of a steady-state solution, a value of x where f(x)=0, and stability (not all steady-state solutions are equal). For problems of the form dx/dt = f(x,μ), we introduce the ideas of a steady-state diagram and bifurcation points. The typical Australian honours student will have seen most, possible all, of these ideas in lower-level courses. At Wollongong all these ideas are taught to first-year undergraduates (MATH111: Applied Mathematical Modelling). Don't worry! More advanced material and new ideas will be covered in latter chapters. This chapter is intended as a gentle introduction to some of the themes of the course.

## Prerequisites

No knowledge of applied mathematics is assumed. Little knowledge above second year calculus is required. You will be at a considerable advantage answering questions on assignments and the final exam if you know some basic commands in a computer algebra system (CAS), such as Maple, Mathematica, or any equivalent package. You will NOT need to write programs using constructs such as loops and if statements. You need to know how to plot functions, differentiate functions and solve equations.

You will not be at a disadvantage if, at the start of the course, you do not know how to use a CAS. However, you will need to learn the basics of a CAS during the first couple of weeks of the course. This will not require a large investment of time.

I can provide some support for students using Maple. I can not provide support for students using an alternative CAS.

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.

## Lecture Notes

Where appropriate I've listed some alternative reading that reinforces the material in each chapter.

### Appendics

• Appendix A. Differential Equations: Definitions.
• Appendix B. Taylor Series.  B.1 Introduction B.2 Taylor series expansion of a function of one variable B.3 Taylor series expansion of a function of two variables
• Appendix C: Invariant Regions.
• Appendix D: Polar co-ordinates.  D.1 Transforming a planar system of differential equations from Cartesian co-ordinates to polar co-ordinates D.2 Things to do.
• Appendix E: Maxima and minima of a function of two variables.  E.1 Stationary points and the test for stationary points E.2 Questions. E.3 Things to do.
• Appendix F: Scaling differential equations.
• Appendix G: Some common models.  G.1 Quadratic autocatalysis.

## Assignments

Note that the chapter numbers have not been the same from year-to-year.

 2008 2009 2010 Week 2 Chapter 1 Week 3 Chapter 1 Chapters 1 & 2 (sections 2.1-2.3) Week 4 Chapter 2 (sections 2.1-2.3) Week 5 Chapter 2 Chapter 2 Chapter 2 (sections 2.4 & 2.5 (sections 2.4 & 2.5 Week 7 Chapter 3 Week 8 Chapter 3 Chapters 3 & 4 Week 9 Chapter 5 Chapter 5 Week 10 Chapter 6 Chapters 3 & 6 Week 11 Chapter 6/TR> Week 12 Chapter 7 Chapters 6 & 8 Week 13 Chapter 8

## Tutorials

 2010 Week 2 Chapter 1 (sections 1.1-1.4) Week 3 Chapter 1 (sections 1.5-1.6) Week 5 Chapter 2 Week 6 Chapter 3 (integrating two coupled first-order differential equations)

## Sample exam papers

 2008 2009

## Maple Code

Chapter One. First-Order Non-Linear Differential Equations
singleode.txt. Integrate a single first-order differential equation.
Chapter Three. Second-order differential equations: Steady-state solutions and their stability
doubleode.txt. Integrate a system of two first-order differential equations.
liapunov.txt. Finding a Lyapunov function that satisfies theorem 3.1. (Can be used to prove that a steady-state solution is either stable or unstable depending upon circumstances).