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 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.

Prerequisites

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
  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
  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.
• 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. A simple bioreactor model
  1. Introduction
  2. Model equations
  3. Analysis of the model with X^*₀=0.
• Chapter Five. Second-order differential equations. The absence of periodic behaviour
  1. Introduction.
  2. Bendixson's Criteria and Dulac's test.
  3. Conclusions.
  4. Maple commands.
  5. Revision of key ideas.
  6. Questions on Bendixson's criteria and Dulac's test.
• 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 mimnima 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.

Assignments

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

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

Tutorials

Sample exam papers

2008

Maple Code

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

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.