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.

1. Introduction.
2. Content.
3. Prerequisites.
4. Assessment.
5. Lecture Notes.
   • Chapters.
   • Appendices.
6. Assignments.
7. Tutorials.
8. Sample exam papers.
9. Maple Code.

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

If you are a student at the University of Wollongong then you should use Maple. If you have not used Maple before then a good place to begin is the following (free) book: http://www.uow.edu.au/content/groups/public/@web/@inf/@math/documents/doc/uow046825.pdf.

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.

Chapters

• Chapter One: First-Order Non-Linear Differential Equations
  1. Introduction
  2. First-order ODEs: Graphical insights
  3. Steady-state solutions
  4. Stability of steady-state solutions
  5. Steady-state diagrams and bifurcation points
  6. The spruce budworm model
  7. Conclusions
  8. Maple commands
  9. Revision of key ideas
  10. Questions on first-order non-linear differential equations
  11. Things to do
  Background reading
  F. Brauer and Castillo-Chávez, C. 2001. Mathematical Models in Population Biology and Epidemiology. Springer-Verlag, Berlin, 1st edition. Chapter 1.4. Qualitative Analysis, pages 17-28.
  For more information on the Spruce Budworm model see
  J.D. Murray. 1989. Mathematical Biology. Springer-Verlag, Berlin, 2nd edition. Chapter 1.2. Insect Outbreak Model: Spruce Budworm, pages 4-8.

  Most of this chapter (1.1-1.5) is based upon notes from a first-year mathematical modelling course.

• Chapter Two: Singularity theory with a distinguished parameter
  1. Introduction.
  2. Singularity theory with a distinguished parameter.
  3. Singularity theory and bifurcation points.
  4. How do steady-state diagrams change in a `fundamental' way?
  5. Constructing a bifurcation diagram.
  6. Conclusion.
  7. Miscellaneous comments. (Things to add into the next set of notes).
  Background reading
  
  1. Balakotaiah, V., and Luss, D. (1982). Structure of the steady-state solutions of lumped-parameter chemically reacting systems. Chemical Engineering Science, 37(11), 1611-1623.
  2. Balakotaiah, V., and Luss, D. (1983). Multiplicity features of reacting systems. Chemical Engineering Science, 38(10), 1709-1721.

• Chapter Three: Second-order differential equations:
  Steady-state solutions and their stability
  1. Introduction.
  2. Steady-state solutions.
  3. Linearised stability of steady-state solutions.
  4. Steady-state diagrams and bifurcation points.
  5. Stability using a Liapunov function.
  6. Conclusions.
  7. Maple commands.
  8. Revision of key ideas.
  9. Questions on steady-states and stability.
  10. Things to do.
• Chapter Four. Second-order differential equations. The absence of periodic behaviour
  1. Introduction.
  2. Bendixson's Criteria and Dulac's test.
  3. Applying Bendixson's Criteria and Dulac's test.
  4. Conclusions.
  5. Maple commands.
  6. Revision of key ideas.
  7. Questions on Bendixson's criteria and Dulac's test.
  8. Things to do.
• Chapter Five. Second-order differential equations: Two simple applications. (29th April 2009. I have revised the PDF but I have not updated the PDF available on this web-page. The new version is only available to students enrolled on the course).
  1. Introduction
  2. A simple bioreactor model
  3. To Do
  4. History
• Chapter Six. Second-order differential equations. Periodic behaviour
  1. Introduction.
  2. Motivation.
  3. The Hopf bifurcation theorem.
  4. The first Liapunov coefficient.
  5. Sub-critical and super-critical Hopf bifurcations.
  6. `Normal form' for the Hopf bifurcation.
  7. Steady-state diagrams.
  8. Conclusions.
  9. Maple commands.
  10. Revision of key ideas
  11. Questions on the Hopf bifurcation.
  12. Things to do
• Chapter Seven. Second-order differential equations: Bifurcations and steady-state diagrams.
  1. Introduction.
  2. Singularity theory and bifurcation points
  3. Steady-state diagrams
  4. Conclusions
  5. Maple commands
  6. Revision of key ideas
  7. Questions on bifurcations and steady-state diagrams.
  8. Things to do
• Chapter Eight. Degenerate Hopf bifurcations.
  1. Introduction.
  2. The double-zero (Bogdanov-Takens) bifurcation
  3. The double Hopf bifurcation.
  4. The Bautin (generalized Hopf) bifurcation
  5. Conclusions
  6. Maple commands
  7. Revision of Key Ideas
  8. Questions on degenerate Hopf bifurcations
  9. Things to do

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
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.

Download the code.

Chapter Three. Second-order differential equations: Steady-state solutions and their stability

doubleode.txt. Integrate a system of two first-order differential equations.

Download the code.

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).

Download the code.

liapunov2.txt Finding a Lyapunov function that satisfies theorem 3.3. (Can be used to prove that a steady-state solution is unstable).

Download the code.

Chapter Six. Second-order differential equations: Periodic behaviour.

hopf-liapunov.txt. Calculate the first liapunov number for a planar system of differential equations.

Download the code.