MATH971: Applied NonLinear Differential Equations
This page contains material for the course: MATH 971
Applied NonLinear 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.
 Content.
 Prerequisites.
 Assessment.
 Lecture Notes.
 Assignments.
 Tutorials.
 Sample exam papers.
 Maple Code.
This course provides an introduction to applied nonlinear
ordinary differential equations. This course is
applied mathematics. There will be no technical lemmas or abstract
definitions!
The course typically consists of twelve
twohour 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.
Topics to be covered include (but are not limited to):
 Firstorder differential equation:
Graphical insights,
steadystate solutions and their stability,
steadystate diagrams and bifurcations.
 Singularity theory with a distinguished parameter:
singularity theory and bifurcation points, constructing
static bifurcation diagrams.
 Systems of two firstorder differential equations
 steadystate solutions and their stability:
local and Liapunov.
 the absence of periodic solutions:
Bendixon's Criteria and Dulac's Test.
 periodic behaviour: the Hopf bifurcation Theorem,
subcritical and supercritical Hopf bifurcations.
 bifurcations and steadystate diagrams:
singularity theory and bifurcation points.
 degenerate Hopf bifurcations:
the double Hopf bifurcation, the Bautin bifurcation, the doublezero
eigenvalue bifurcation.
In Chapter One we consider
a single autonomous firstorder 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
steadystate solution, a value of x where
f(x)=0, and stability (not all steadystate solutions
are equal). For problems of the form
dx/dt = f(x,μ), we introduce the ideas of
a steadystate diagram and bifurcation points.
The typical Australian honours student will have seen
most, possible all, of these ideas in lowerlevel courses. At Wollongong
all these ideas are taught to firstyear 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.
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.
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.
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.
Where appropriate I've listed some alternative reading that
reinforces the material in each chapter.

 Introduction
 Firstorder ODEs: Graphical insights
 Steadystate solutions
 Stability of steadystate solutions
 Steadystate diagrams and bifurcation points
 The spruce budworm model
 Conclusions
 Maple commands
 Revision of key ideas
 Questions on firstorder nonlinear differential equations
 Things to do
Background reading
F. Brauer and CastilloChávez, C. 2001.
Mathematical Models in Population Biology and Epidemiology.
SpringerVerlag, Berlin, 1st edition.
Chapter 1.4. Qualitative Analysis, pages 1728.
For more information on the Spruce Budworm model see
J.D. Murray. 1989. Mathematical Biology.
SpringerVerlag, Berlin, 2nd edition.
Chapter 1.2. Insect Outbreak Model: Spruce Budworm, pages 48.
Most of this chapter (1.11.5) is based upon notes from a firstyear
mathematical modelling course.
 Chapter Two: Singularity theory with a distinguished parameter
 Introduction.
 Singularity theory with a distinguished parameter.
 Singularity theory and bifurcation points.
 How do steadystate diagrams change in a `fundamental'
way?
 Constructing a bifurcation diagram.
 Conclusion.
 Miscellaneous comments. (Things to add into the next set
of notes).
Background reading
 Balakotaiah, V., and Luss, D. (1982). Structure of the steadystate
solutions of lumpedparameter chemically reacting systems.
Chemical Engineering Science, 37(11),
16111623.
 Balakotaiah, V., and Luss, D. (1983). Multiplicity
features of reacting systems.
Chemical Engineering Science, 38(10),
17091721.
 Chapter Three: Secondorder differential equations:
Steadystate solutions and their stability
 Introduction.
 Steadystate solutions.
 Linearised stability of steadystate solutions.
 Steadystate diagrams and bifurcation points.
 Stability using a Liapunov function.
 Conclusions.
 Maple commands.
 Revision of key ideas.
 Questions on steadystates and stability.
 Things to do.
 Chapter Four. Secondorder differential equations.
The absence of periodic behaviour
 Introduction.
 Bendixson's Criteria and Dulac's test.
 Applying Bendixson's Criteria and Dulac's test.
 Conclusions.
 Maple commands.
 Revision of key ideas.
 Questions on Bendixson's criteria and Dulac's test.
 Things to do.
 Chapter Five. Secondorder differential equations:
Two simple applications.
(29th April 2009. I have revised the PDF but I have not updated the
PDF available on this webpage. The new version is only available
to students enrolled on the course).
 Introduction
 A simple bioreactor model
 To Do
 History
 Chapter Six. Secondorder differential equations.
Periodic behaviour
 Introduction.
 Motivation.
 The Hopf bifurcation theorem.
 The first Liapunov coefficient.
 Subcritical and supercritical Hopf bifurcations.
 `Normal form' for the Hopf bifurcation.
 Steadystate diagrams.
 Conclusions.
 Maple commands.
 Revision of key ideas
 Questions on the Hopf bifurcation.
 Things to do
 Chapter Seven.
Secondorder differential equations:
Bifurcations and steadystate diagrams.
 Introduction.
 Singularity theory and bifurcation points
 Steadystate diagrams
 Conclusions
 Maple commands
 Revision of key ideas
 Questions on bifurcations and steadystate diagrams.
 Things to do

 Chapter Eight.
Degenerate Hopf bifurcations.
 Introduction.
 The doublezero (BogdanovTakens) bifurcation
 The double Hopf bifurcation.
 The Bautin (generalized Hopf) bifurcation
 Conclusions
 Maple commands
 Revision of Key Ideas
 Questions on degenerate Hopf bifurcations
 Things to do
 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 coordinates.
D.1  Transforming a planar system of differential equations
from Cartesian coordinates to polar coordinates 
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.

Note that the chapter numbers have not been the same from
yeartoyear.
 2008 
2009 
2010 
Week 2 
Chapter 1 
Week 3  
Chapter 1 
Chapters 1 & 2

   (sections 2.12.3) 
Week 4 
Chapter 2 
 (sections 2.12.3) 
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 
2010 
Week 2 
Chapter 1 (sections 1.11.4) 
Week 3 
Chapter 1 (sections 1.51.6) 
Week 5 
Chapter 2 
Week 6 
Chapter 3 (integrating two coupled firstorder
differential equations) 
 Chapter One. FirstOrder NonLinear Differential Equations
 singleode.txt. Integrate a single
firstorder differential equation.
 Download the code.
 Chapter Three. Secondorder differential equations:
Steadystate solutions and their stability
 doubleode.txt.
Integrate a system of two firstorder differential equations.
 Download the code.
 liapunov.txt.
Finding a Lyapunov function that satisfies theorem 3.1.
(Can be used to prove that a steadystate 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 steadystate solution is
unstable).
 Download the code.
 Chapter Six. Secondorder differential equations: Periodic behaviour.
 hopfliapunov.txt.
Calculate the first liapunov number for a planar system of differential
equations.
 Download the code.
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Page Created: 23rd March 2005.
Last Updated: 5th June 2012.