Introduction

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.

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 are answered by the Picard Theorem on existence and uniqueness of solutions of ODEs, and its consequences.

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 qualitative behaviour of solutions even when we cannot solve the equations exactly. We will develop techniques to answer important questions about the stability/attraction properties (or instabilities) of given solutions, often fixed points.

We will eventually apply these powerful methods to particular examples of practical importance, including the Lotka-Volterra model for the competition between two species, Hamiltonian systems, and the Lorenz equations, and give an informal introduction to some more advanced topics (e.g. bifurcation theory, Lyapunov exponents).

Module page

Moodle - contains lecture notes

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, 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

Part II: Qualitative Theory of Initial Value Problems

1) Stability: linear stability, Lyapunov stability, convergence to equilibrium

2) Qualitative Theory in R^2: phase plane analysis, equilibria, local phase portraits (sketch of Hartmann‐Grobman Thm), limit cycles, attractors, Bendixson‐Dulac, Poincare‐Bendixson)

3) Informal introduction to chaos, bifurcation, catastrophe to motivate further modules in dynamical systems (definitions and relation to applications detailed above).

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.