Unit information: Applied dynamical systems in 2015/16

Unit name	Applied dynamical systems
Unit code	MATHM0010
Credit points	10
Level of study	M/7
Teaching block(s)	Teaching Block 2C (weeks 13 - 18)
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

Unit aims

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.

General Description of the Unit

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.

NOTE: This unit is also part of the Oxford-led Taught Course Centre (TCC), and is taken by first- and second-year PhD students in Bristol and its TCC partner departments. The unit has been designed primarily with a postgraduate audience in mind. Undergraduate students should not normally take more than one TCC unit per semester.

Relation to Other Units

This is intended as a standalone course for students with the relevant prerequisites. A complementary (pure mathematics) perspective is given in Dynamical Systems and Ergodic Theory, and more detail on bifurcation analysis may be found in the Engineering unit Nonlinear Dynamics and Chaos. Connections may also be made with Statistical Mechanics and with applied probability units.

Further information is available on the School of Mathematics website: http://www.maths.bris.ac.uk/study/undergrad/

Intended Learning Outcomes

Learning Objectives

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 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.

Transferable Skills

Mathematical modelling, computational, written and oral communication skills.

Teaching Information

The unit will be delivered through lectures as well as written and computational homework problems.

Assessment Information

60% Examination and 40% Coursework.

Raw scores on the examinations will be determined according to the marking scheme written on the examination paper. The marking scheme, indicating the maximum score per question, is a guide to the relative weighting of the questions. Raw scores are moderated as described in the Undergraduate Handbook.

Reading and References

• J. C. Sprott, Chaos and time series analysis, OUP 2003.

• R. L. Devaney, An Introduction to chaotic dynamical systems, Westview Press, 2003.

• B. Hasselblatt and A. Katok, A first course in dynamics, CUP 2003.