MA25412 Theory of ODEs
Introductory description
Many fundamental problems in the applied sciences reduce to understanding solutions of ordinary differential equations (ODEs). Examples include the laws of Newtonian mechanics, predatorprey models in Biology, and nonlinear oscillations in electrical circuits, to name only a few. These equations are often too complicated to solve exactly, so one tries to understand qualitative features of solutions.
Some questions we will address in this course include:
When do solutions of ODEs exist and when are they unique? What is the long time behaviour of solutions and can they "blowup" in finite time? These questions culminate in the famous PicardLindelof theorem on existence and uniqueness of solutions of ODEs.
The main part of the course will focus on phase space methods. This is a beautiful geometrical approach which often enables one to understand the behaviour of solutions near critical points  often exactly the regions one is interested in. Different trajectories will be classified and we will develop techniques to answer important questions on the stability properties (or lack thereof) of given solutions.
We will eventually apply these powerful methods to particular examples of practical importance, including the LotkaVolterra model for the competition between two species and to the Van der Pol and Lienard systems of electrical circuits.
The course will end with a discussion of the SturmLiouville theory for solving boundary value problems.
Module aims
Extend the knowledge of first year ODEs with a mixture of applications, modelling and theory to prepare for more advanced modules later on in the course.
Outline syllabus
This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.
Introduction: The module will begin with the introduction of a few model systems to motivate questions and techniques; which will reappear throughout the module, applying the new techniques as they are acquired. Examples: Lotka‐Volterra, Duffing, Lorenz, Hodgkin‐Huxley and Fitzhugh‐Nagumo, general Hamiltonian systems / nonlinear oscillator, general gradient flows.
Part I: Theory of Initial Value Problems
 Picard Thm in R^n: concept of well‐posedness, local existence and uniqueness,
non‐uniqueness, maximal existence interval, blowup  Linear theory in R^n: general solutions for constant coefficients, exponential of a
matrix, variation of constants in R^n, Gronwall Lemma  Euler's Method: convergence, long‐time behaviour
Part II: Qualitative Theory of Initial Value Problems
4. Stability: linear stability, Lyapunov stability, convergence to equilibrium
5. Qualitative Theory in R^2: phase plane analysis, equilibria, local phase portraits (sketch
of Hartmann‐Grobman Thm), limit cycles, attractors, Bendixson‐Dulac, Poincare‐Bendixson,
6. Informal introduction to chaos, bifurcation, catastrophe to motivate further modules
in dynamical systems (definitions and relation to applications detailed above).
Part III: Theory of Linear Boundary Value Problems (time permitting)
7. Linear 2‐point bvps: Fredholm alternative and Green's function, link to Fourier series 8. Sturm‐Liouville theory
Learning outcomes
By the end of the module, students should be able to:
 Determine the fundamental properties of solutions to certain classes of ODEs, such as existence and uniqueness of solutions.
 Sketch the phase portrait of 2dimensional systems of ODEs and classify critical points and trajectories.
 Classify various types of orbits and possible behaviour of general nonlinear ODEs.
 Understand the behaviour of solutions near a critical point and how to apply linearization techniques to a nonlinear problem.
 Apply these methods to certain physical or biological systems.
Indicative reading list
Elementary Differential Equations and Boundary Value Problems, Boyce DiPrima 1997
Differential Equations, Dynamical Systems, and an Introduction to Chaos, Hirsch, Smale 2003
Nonlinear Systems, Drazin 1992
Subject specific skills
See learning outcomes
Transferable skills
Students will acquire key reasoning and problem solving skills which will empower them to address new problems with confidence.
Study time
Type  Required 

Lectures  30 sessions of 1 hour (25%) 
Seminars  10 sessions of 1 hour (8%) 
Private study  80 hours (67%) 
Total  120 hours 
Private study description
Private study, preparation, revision for exams, reviewing lectured material and working on set exercises.
Costs
No further costs have been identified for this module.
You do not need to pass all assessment components to pass the module.
Students can register for this module without taking any assessment.
Assessment group B1
Weighting  Study time  

Inperson Examination  100%  

Assessment group R
Weighting  Study time  

Inperson Examination  Resit  100%  

Feedback on assessment
Marked assignments and exam feedback.
Courses
This module is Optional for:
 Year 3 of UMAAGL11 Undergraduate Mathematics and Economics
 Year 2 of USTAG1G3 Undergraduate Mathematics and Statistics (BSc MMathStat)
 Year 2 of USTAGG14 Undergraduate Mathematics and Statistics (BSc)
 Year 2 of USTAY602 Undergraduate Mathematics,Operational Research,Statistics and Economics
This module is Core option list B for:
 Year 3 of UMAAGV17 Undergraduate Mathematics and Philosophy

UMAAGV19 Undergraduate Mathematics and Philosophy with Specialism in Logic and Foundations
 Year 3 of GV19 Mathematics and Philosophy with Specialism in Logic and Foundations
 Year 3 of GV19 Mathematics and Philosophy with Specialism in Logic and Foundations
This module is Core option list C for:
 Year 2 of UMAAGV19 Undergraduate Mathematics and Philosophy with Specialism in Logic and Foundations
This module is Core option list D for:
 Year 4 of UMAAGV19 Undergraduate Mathematics and Philosophy with Specialism in Logic and Foundations
This module is Option list A for:

UMAAG105 Undergraduate Master of Mathematics (with Intercalated Year)
 Year 2 of G105 Mathematics (MMath) with Intercalated Year
 Year 3 of G105 Mathematics (MMath) with Intercalated Year
 Year 2 of USTAG300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics

UMAAG100 Undergraduate Mathematics (BSc)
 Year 2 of G100 Mathematics
 Year 3 of G100 Mathematics

UMAAG103 Undergraduate Mathematics (MMath)
 Year 2 of G103 Mathematics (MMath)
 Year 3 of G103 Mathematics (MMath)

UMAAG106 Undergraduate Mathematics (MMath) with Study in Europe
 Year 2 of G106 Mathematics (MMath) with Study in Europe
 Year 3 of G106 Mathematics (MMath) with Study in Europe
 Year 2 of UMAAG1NC Undergraduate Mathematics and Business Studies
 Year 2 of UMAAG1N2 Undergraduate Mathematics and Business Studies (with Intercalated Year)
 Year 2 of UMAAGL11 Undergraduate Mathematics and Economics
 Year 2 of UECAGL12 Undergraduate Mathematics and Economics (with Intercalated Year)
 Year 2 of UPXAFG33 Undergraduate Mathematics and Physics (BSc MMathPhys)
 Year 2 of UPXAGF13 Undergraduate Mathematics and Physics (BSc)
 Year 2 of UPXAFG31 Undergraduate Mathematics and Physics (MMathPhys)

UMAAG101 Undergraduate Mathematics with Intercalated Year
 Year 2 of G101 Mathematics with Intercalated Year
 Year 4 of G101 Mathematics with Intercalated Year
 Year 2 of USTAY602 Undergraduate Mathematics,Operational Research,Statistics and Economics
This module is Option list B for:
 Year 2 of UCSAG4G1 Undergraduate Discrete Mathematics
 Year 2 of UCSAG4G3 Undergraduate Discrete Mathematics
 Year 3 of USTAG1G3 Undergraduate Mathematics and Statistics (BSc MMathStat)
 Year 4 of USTAG1G4 Undergraduate Mathematics and Statistics (BSc MMathStat) (with Intercalated Year)
 Year 3 of USTAGG14 Undergraduate Mathematics and Statistics (BSc)
 Year 4 of USTAGG17 Undergraduate Mathematics and Statistics (with Intercalated Year)