## MATH2391 Nonlinear Differential Equations

### 10 creditsClass Size: 170

Module manager: Professor Allan Fordy
Email: A.P.Fordy@leeds.ac.uk

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2020/21

### Pre-requisite qualifications

MATH1005 or (MATH1010 and MATH1012) or (MATH1400 and MATH1060) or (MATH1400 and MATH1331) or (PHYS1300 and MATH1060), or equivalent.

This module is approved as a discovery module

### 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 Workshop 10 1.00 10.00 Lecture 11 1.00 11.00 Private study hours 79.00 Total Contact hours 21.00 Total hours (100hr per 10 credits) 100.00

### Private study

Regular examples sheets

### Opportunities for Formative Feedback

Regular problem solving assignments

### 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 . 15.00 Total percentage (Assessment Coursework) 15.00

There is no resit available for the coursework component of this module. If the module is failed, the coursework mark will be carried forward and added to the resit exam mark with the same weighting as listed above.

Exams
 Exam type Exam duration % of formal assessment Open Book exam 2 hr 00 mins 85.00 Total percentage (Assessment Exams) 85.00

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