Dynamical Systems (MSc) - MAT00062H

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This module is for postgraduate students only.

Module will run

Occurrence	Teaching period
A	Autumn Term 2022-23

Module aims

Dynamical systems describe the time evolution of systems which arise from physics, biology, chemistry 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

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 and graduate skills

• 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.

Indicative assessment

Task	% of module mark
Closed/in-person Exam (Centrally scheduled)	100

Special assessment rules

Pass/fail

Additional assessment information

40% of the final exam mark comes from coursework

Indicative reassessment

Task	% of module mark
Closed/in-person Exam (Centrally scheduled)	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.

Indicative reading

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