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2005/06 Undergraduate Module Catalogue

MATH2391 Nonlinear Differential Equations

10 creditsClass Size: 200

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2005/06

Pre-requisites

MATH1932 or (MATH1960 and MATH1970) or MATH1400 or MATH2450 or equivalent. (MATH2360 or MATH2420 Recommended but not essential)

This module is approved as an Elective

Objectives

On completion of this module, students should be able to:

a) sketch phase plane portraits of second-order linear and nonlinear ODEs
b) sketch bifurcation diagrams and identify bifurcation points
c) determine the stability of equilibrium points using a variety of methods
d) determine the exsistence or otherwise of periodic orbits in second order autonomous nonlinear ODEs using Dulac's criterion, Lyapunov functions and the Poincare-Bendixson Theorem.

Syllabus

1. Existence and uniqueness of ordinary differential equations. Examples of finite time blow-up abd non-uniqueness of solutions.

2. First order nonlinear ODEs. Stability of equilibrium solutions. Interpretation of the nonlinear ODE as a vector field.

3. Bifurcation theory for first order nonlinear ODEs: the saddle-node, transcritical and pitchfork bifurcations. Discussion of structural stability.

4. Second order nonlinear ODEs. Phase portraits. Construction of the exponential matrix, including Jordan canonical form for 2 x 2 matrices.

5. Second order nonlinear ODEs. Equilibrium solutions and linear stability theory. Using MAPLE to assist drawing phase portraits.

6. Elementary theory of periodic orbits. Dulac's criterion, Lyapunov functions, Poincare index theory, Poincare-Bendixson Theorem.

7. Bifurcations in second order nonlinear ODEs: the Hopf bifurcation., Only treated as a statement. without proof or extended study.

Teaching methods

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

Lectures 22 hours


Example Classes 11 hours

Private study

Regular examples sheets

Methods of assessment

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

80% 2 hour written examination at the end of the semester
20% coursework

Reading list

The reading list is available from the Library website

Last updated: 13/05/2005

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