## 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 2022/23

### 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 do the following (where appropriate) for first and second order linear and nonlinear ODEs:

a) sketch phase portraits;
b) determine the stability of equilibrium points via the eigenvalues of its Jacobian;
c) sketch bifurcation diagrams, identify bifurcation points and classify fold (saddle-node), transcritical and pitchfork bifurcations;

### Syllabus

1. Existence and uniqueness of ordinary differential equations. Examples of finite time blow-up and 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.
4. Second order linear ODEs. Phase portraits. Construction of the exponential matrix, including Jordan canonical form for 2 x 2 matrices.
5. Second order nonlinear ODEs. Equilibrium solutions, linear stability theory and drawing phase portraits.
6. Additional topics (at the module leader's discretion): first integrals, theory of periodic orbits; perturbation approaches, computational methods.

### Teaching methods

 Delivery type Number Length hours Student hours Workshop 10 1.00 10.00 Lecture 22 1.00 22.00 Private study hours 68.00 Total Contact hours 32.00 Total hours (100hr per 10 credits) 100.00

### Private study

Regular examples sheets

### Opportunities for Formative Feedback

Regular problem solving assignments

### Methods of assessment

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 Standard exam (closed essays, MCQs etc) 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