MAT-30002 - Non-linear Differential Equations
Coordinator: Danila Prikazchikov Tel: +44 1782 7 33414
Lecture Time: See Timetable...
Level: Level 6
Credits: 15
Study Hours: 150
School Office: 01782 733075

Programme/Approved Electives for 2022/23

None

Available as a Free Standing Elective

No

Co-requisites

None

Prerequisites

MAT-20008 Differential Equations

Barred Combinations

None

Description for 2022/23

Systems that evolve in time can often be modelled by differential equations. There are countless examples of such systems from the physical world including the weather, climate change, stock markets, the economy, population dynamics, mechanical systems, etc. The great variety of behaviours exhibited by these systems is reflected in the solutions to the corresponding differential equations. This module introduces a number of methods for identifying and classifying various types of behaviour in various types of differential equation. While linear differential equations model some processes, the majority are described by nonlinear equations, and it is these that display the greatest diversity of behaviour. However, very few nonlinear differential equations have exact solutions. Nevertheless, a great deal of insight can be obtained from qualitative methods. This module focuses on geometric methods for constructing phase plane representations of dynamics and perturbation methods for obtaining approximate solutions. With these tools it is then possible to examine the changes in behaviour that can occur when a parameter is varied, and bifurcation theory is introduced to describe this. The relation between the evolution of differential equations and the evolution of maps is explained, and more exotic behaviour, like period doubling and chaos, are then studied in terms of the dynamics of maps.

Aims
The aim of this module is to study the qualitative behaviour of solutions to ordinary differential equations through the use of phase plane analysis.

Intended Learning Outcomes

analyse the qualitative behaviour of solutions to ordinary differential equations through the use of the phase plane: 1,3
demonstrate knowledge of the theory of steady bifurcations: 2,3
demonstrate knowledge of the theory of self-excited oscillations, their representation by a limit cycle, and the generation of limit cycles at change in stability of a fixed point: the Hopf bifurcatio: 3
demonstrate knowledge of the theory of forced oscillations and the use of perturbation methods to discover the effect of nonlinearity in producing harmonics and the consequences for resonance: 3
demonstrate knowledge of the theory and application of Poincaré Maps: 3
construct the phase plane by locating and classifying fixed points: 1,3

Study hours

Lectures and examples classes: 36 hours
Independent study: 112 hours
Online examination : 2 hours

School Rules

None

Description of Module Assessment

1: Assignment weighted 15%
Take-home assignment
Online assignment will reflect on the knowledge gained in the first three chapters (weeks 1-6). Students are expected to spend at least 6 hours in revision and solving the assignment.

2: Class Test weighted 15%
40-minute class test
A 40-minute class test will rely on the material of chapter 4.

3: Unseen Exam weighted 70%
2-hour unseen exam
The examination paper will consist of no less than five and not more than eight questions all of which are compulsory.