## MATH2391 Nonlinear Differential Equations

### 10 creditsClass Size: 200

Module manager: Professor A.V. Mikhailov
Email: sashamik@maths.leeds.ac.uk

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2008/09

### Pre-requisite qualifications

MATH1970 or MATH1932 or MATH1960 or MATH1400 or MATH2450, or equivalent. MATH2365 and MATH2420 are recommended but not essential.

### Co-requisites

 MATH2015 Analysis 2

This module is approved as an Elective

### Module summary

Nonlinear systems occur widely in the real world, and may produce oscillations or even wild chaotic fluctuations even when influenced by a constant external force. This course provides a first introduction to the mathematics behind such behaviour.

### 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

 Delivery type Number Length hours Student hours Example Class 9 1.00 9.00 Lecture 22 1.00 22.00 Private study hours 69.00 Total Contact hours 31.00 Total hours (100hr per 10 credits) 100.00

### 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

Coursework
 Assessment type Notes % of formal assessment In-course Assessment . 10.00 In-course MCQ . 10.00 Total percentage (Assessment Coursework) 20.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 2 hr 80.00 Total percentage (Assessment Exams) 80.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated