Breadcrumb

Applied dynamical systems (MATHM0010)

Academic Year:

Contents of this document:

Administrative Information

  1. Unit number and title: MATHM0010 Applied dynamical systems
  2. Level: M/7
  3. Credit point value: 10 credit points
  4. Year: 12/13
  5. First Given in this form: 2012
  6. Unit Organiser: Carl Dettmann
  7. Lecturer: Dr Carl Dettmann
  8. Teaching block: 2
  9. Prerequisites: 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.

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 units (Advanced) Nonlinear Dynamics and Chaos.  Connections may also be made with Statistical Mechanics and with applied probability units.

Teaching Methods

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

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 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.

Assessment Methods

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.

Award of Credit Points

Credit points for the unit are gained by passing the unit (i.e. getting a final assessment mark of 50 or over).

Transferable Skills

Mathematical modelling, computational, written and oral communication skills.

Texts

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

Syllabus

Approximate numbers of lectures are indicated; numerical methods will be emphasised throughout.

Motivating examples, maps, flows, Poincare and time-one maps (1).

Fixed points, periodic orbits, local stability analysis and bifurcations (3)

Expanding and hyperbolic maps, Markov partitions, symbolic dynamics, Lyapunov exponents, stable and unstable manifolds, Cantor sets and dimensions (4).

Ergodic theory: Poincare recurrence, Birkoff ergodic theorem, ergodicity, mixing, metric entropy (2).

Integrable Hamiltonian systems and perturbations, standard map and Chirikov criterion, global bifurcations (2).

Intermittency, Pomeau-Manneville maps, introduction to infinite ergodic theory (3)