Unit information: Applied dynamical systems in 2013/14

Unit name	Applied dynamical systems
Unit code	MATHM0010
Credit points	10
Level of study	M/7
Teaching block(s)	Teaching Block 2 (weeks 13 - 24)
Unit director	Professor. Dettmann
Open unit status	Not open
Pre-requisites	MATH11005 (Linear Algebra and Geometry), MATH11006 (Analysis 1), MATH 20101 (Ordinary Differential Equations), MATH 20700 (Numerical Analysis). MATH36206 or MATHM6206 (Dynamical Systems and Ergodic Theory) is helpful but optional. Students will be expected to have attained a degree of mathematical maturity and facility at least to the standard of a beginning Level M/7 student.
Co-requisites	None
School/department	School of Mathematics
Faculty	Faculty of Science

Description including Unit Aims

This unit provides an introduction to dynamical systems from an applied mathematics point of view, surveying the main areas of the subject, with an emphasis on concepts and on analytical and numerical methods that form a foundation for research in applied mathematics and theoretical physics. Systems considered range from almost regular through intermittent to strongly chaotic. Relevant geometrical structures such as bifurcation diagrams, fractal attractors and repellers are discussed at the relevant points. While the unit is self-contained, it is advantageous to first complete Dynamical Systems and Ergodic Theory, available at level H/6 or M/7, which emphasises hyperbolic and ergodic dynamics from a pure mathematics perspective.

The aims of this unit are:

• To inspire students with the unity, the richness and variety of dynamical systems,
• To prepare students to solve research problems involving dynamics by recognising pertinent concepts, where needed undertaking further self-directed reading, identifying possible strategies, and proceeding to implement them,
• To develop confidence with relevant numerical techniques

Intended Learning Outcomes

A student completing this unit successfully will be able to:

• Locate and analyse the stability of fixed points and periodic orbits of maps and flows;
• Identify commonly encountered local and global bifurcations;
• Quantify piecewise linear expanding and hyperbolic dynamics and associated sets, and apply their understanding to a qualitative treatment of more general hyperbolic systems;
• Be familiar with the main ergodic properties of dynamical systems, logical connections, known results and conjectures;
• Define integrability of Hamiltonian systems, and give a qualitative and semi-quantitative analysis of perturbed integrable dynamics;
• Identify sources of intermittency in dynamical systems, synthesising and contrasting understanding from earlier sections of the unit;
• Identify applications of each of the main classes of dynamical systems, stating features of their long time behaviour;
• Accurately simulate and quantify dynamical systems numerically, assessing likely sources of uncertainty.

Teaching Information

The unit will be delivered through lectures. The lectures will be transmitted over the internet as part of the Taught Course Centre (TCC). The TCC is a consortium of five mathematics departments, including Bath, Bristol, Imperial College, Oxford and Warwick.

Assessment Information

Formative homework exercises will be assigned throughout the unit, both theoretical and numerical.

The final assessment mark will be based on:

• a 1-hour written examination (60%)
• a project of 2000 words (30%)
• a 5 minute presentation (10%)

where the requirements for the project and the presentation will include a brief literature survey, and analytical and numerical investigations of a dynamical system.

Reading and References

• J. C. Sprott, Chaos and time series analysis, OUP 2003.
• B. Hasselblatt and A. Katok, A first course in dynamics, CUP 2003.