# Dynamical Systems - MAT00011H

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• Department: Mathematics
• Module co-ordinator: Dr. Jamie Wood
• Credit value: 10 credits
• Credit level: H
• Academic year of delivery: 2019-20

## Module will run

Occurrence Teaching cycle
A Autumn Term 2019-20

## Module aims

Dynamical systems describe the time evolution of systems which arise from physics, biology, chemsitry and other areas. As mathematical objects they are ordinary differential equations, usually nonlinear and therefore not usually able to be explicitly solved. The aim of the course is to see how to make a qualitative analysis of a dynamical system using many different analytic tools. By the end of the course students should be able to analyse planar systems to understand their global dynamics and how these might change as parameters of the system are varied

## Module learning outcomes

• To introduce students to the basic mathematical skills for the qualitative solving of low dimensional systems of ordinary differential equations in continuous time, including dimensionless forms, phase portraits, and bifurcations
• To provide a brief introduction to the way ordinary differential equations can be used to model, explain and interpret real world problems.
• To provide a brief introduction to the theory and concepts that underpin the field of dynamical systems

## Module content

[Pre-requisite modules for Natural Sciences students: Maths for Sciences 1, 2 & 3.]

Subject content

The module aims to introduce key methodological techniques illustrated by examples, working up from low dimensions to implications in higher dimensions. The course will feature a mixture of traditional lectures complemented by responseware walkthroughs of key selected examples.

Lecture content is as follows

L1 Flows on a line: 1D Equations and exact solutions, dimensionless form (briefly)

L2 Flows on a line: 1D Equations and fixed points and stability,

L3 Bifurcations in 1D Normal Forms. Fold Bifurcations, Transcritical Bifurcations, Pitchfork Bifurcation (Briefly)

W4 The Spruce Budworm Model

L5 Flows in two dimensions: Linear systems, classification

L6 Flows in two dimensions: Dimensionless forms,

L7 Flows in two dimensions: Linearisation for non linear systems, Phase portraits

W8 Building a phase portrait

L9 Introduction to Mathematical Ecology

W10 Lotke-Volterra competition and predation models.

L11 Conservative Systems

W12 Pendulums and Oscillators

L13 Bifurcations revisited, introducing Hopf bifurcations

L14 Limit cycles, Poincare-Bendixon and the Hopf Bifurcation theorem

L15 Introduction to Mathematical Systems Biology

W16 The Brusselator

W17 Predation model with oscillations - heteroclinic bifurcations

L18 Higher dimensions: the Lorenz system and Chaos

(Where L indicates a traditional lecture and W indicates a worked example with interactive responseware.)

• Academic skills: by the end of the module, students should be able to confidently analyse a small system of ordinary differential equations and produce a quantitatively accurate local map and a qualitatively accurate global phase portrait. They should be able to understand how the mathematical changes from bifurcations change the structure of the phase portrait. Some
• Graduate skills: through lectures, examples, classes, students should develop their ability to assimilate, process and engage with new material quickly and efficiently. Students should develop problem solving-skills and learn how to apply techniques to unseen problems as well as demonstrate understanding of some well-established problems in the field.

## Assessment

Task Length % of module mark
University - closed examination
Dynamical Systems
2 hours 100

None

### Reassessment

Task Length % of module mark
University - closed examination
Dynamical Systems
2 hours 100

## Module feedback

Current Department policy on feedback is available in the undergraduate student handbook. Coursework and examinations will be marked and returned in accordance with this policy.

S H Strogatz, Nonlinear Dynamics and Chaos. Westview Press (Perseus), 1994 (York Library Code S7.38 STR)

The information on this page is indicative of the module that is currently on offer. The University is constantly exploring ways to enhance and improve its degree programmes and therefore reserves the right to make variations to the content and method of delivery of modules, and to discontinue modules, if such action is reasonably considered to be necessary by the University. Where appropriate, the University will notify and consult with affected students in advance about any changes that are required in line with the University's policy on the Approval of Modifications to Existing Taught Programmes of Study.