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.
- Content.
- Prerequisites.
- Assessment.
- Lecture Notes.
- Assignments.
- Tutorials.
- Sample exam papers.
- Maple Code.
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.
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
- steady-state 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,
sub-critical and super-critical Hopf bifurcations.
- bifurcations and steady-state diagrams:
singularity theory and bifurcation points.
- 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.
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
- First-order ODEs: Graphical insights
- Steady-state solutions
- Stability of steady-state solutions
- Steady-state diagrams and bifurcation points
- The spruce budworm model
- Conclusions
- Maple commands
- Revision of key ideas
- Questions on first-order non-linear differential equations
- 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
- Introduction.
- Singularity theory with a distinguished parameter.
- Singularity theory and bifurcation points.
- How do steady-state 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 steady-state
solutions of lumped-parameter chemically reacting systems.
Chemical Engineering Science, 37(11),
1611-1623.
- 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
- Introduction.
- Steady-state solutions.
- Linearised stability of steady-state solutions.
- Steady-state diagrams and bifurcation points.
- Stability using a Liapunov function.
- Conclusions.
- Maple commands.
- Revision of key ideas.
- Questions on steady-states and stability.
- Things to do.
- Chapter Four. Second-order 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. 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).
- Introduction
- A simple bioreactor model
- To Do
- History
- Chapter Six. Second-order differential equations.
Periodic behaviour
- Introduction.
- Motivation.
- The Hopf bifurcation theorem.
- The first Liapunov coefficient.
- Sub-critical and super-critical Hopf bifurcations.
- `Normal form' for the Hopf bifurcation.
- Steady-state diagrams.
- Conclusions.
- Maple commands.
- Revision of key ideas
- Questions on the Hopf bifurcation.
- Things to do
- Chapter Seven.
Second-order differential equations:
Bifurcations and steady-state diagrams.
- Introduction.
- Singularity theory and bifurcation points
- Steady-state diagrams
- Conclusions
- Maple commands
- Revision of key ideas
- Questions on bifurcations and steady-state diagrams.
- Things to do
-
- Chapter Eight.
Degenerate Hopf bifurcations.
- Introduction.
- The double-zero (Bogdanov-Takens) 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 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.
|
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 |
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) |
- 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.
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Page Created: 23rd March 2005.
Last Updated: 5th June 2012.