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, predator-prey models in Biology, and non-linear 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 "blow-up" in finite time? These questions culminate in the famous Picard-Lindelof 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 Lotka-Volterra 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 Sturm-Liouville theory for solving boundary value problems.

Module web page

Module aims

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

1. Picard Thm in R^n: concept of well‐posedness, local existence and uniqueness,
   non‐uniqueness, maximal existence interval, blowup
2. Linear theory in R^n: general solutions for constant coefficients, exponential of a
   matrix, variation of constants in R^n, Gronwall Lemma
3. 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 2-dimensional systems of ODEs and classify critical points and trajectories.
• Classify various types of orbits and possible behaviour of general non-linear ODEs.
• Understand the behaviour of solutions near a critical point and how to apply linearization techniques to a non-linear 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.