MATH30031 Bifurcation and Dynamical Systems

This is former 254/MA3031

 SEMESTER: Second
CONTACT: Dr James Montaldi (M/N8) CREDIT RATING: 10
Aims: To introduce nonlinear discrete time dynamical systems and study some of their properties, in particular the kinds of dynamics they can exhibit.
Intended Learning Outcomes: On successful completion of the course students will:
  • Have acquired a basic understanding of discrete time dynamical systems on the interval,
  • Be able to find the fixed and periodic points of simple dynamical systems on the interval, and determine their stability,
  • Have some familiarity with some of the simpler bifurcations that fixed and periodic points can undergo.
  • Have some familiarity with the notion of self-similar fractals, and how they arise as attractors
Pre-requisites: 111,115,151, 155 (211 and 251 are desirable) (ex-UMIST)
MT1121, MT1222, MT2202 (MT2222 is desirable) (ex-VUM)
Dependent Courses: None (desirable for 421).
Course Description: This course introduces discrete time dynamical systems (iterated mappings) and analyses them using the sort of qualitative approaches developed for continuous time systems in 155 (Applied IB).  Mappings of the interval [0, 1] to itself are studied in detail; these are simple examples of discrete time systems but they can show remarkably complex dynamical behaviour, including chaotic dynamics.  The existence of fixed points and periodic points is explored, and the way these change as the system changes (bifurcation theory) is investigated.  The basic ideas of symbolic dynamics as a way of analysing dynamical systems is introduced, and the method is used to show some simple maps have chaotic behaviour.
Teaching Mode: 2 Lectures per week
1 Tutorial per week
Private Study: 5 hours per week
Recommended Text: KT Alligood, TD Sauer, JA Yorke, Chaos - an introduction to dynamical systems,1996, Springer-Verlag
Assessment Methods: Coursework: 20%
Coursework Mode: Problem sheet handed out in Week 5, solutions to be handed in in Week 7.
Examination: 80%
Examination is of 2 hours duration at the end of the Second Semester.
No of lectures: Syllabus
3 Introduction to discrete time systems.  Simple examples and applications.  Fixed and periodic points. Stability.
6 One dimensional systems.  Graphical analysis.   Stability of fixed and periodic points.   Sensitive dependence on initial conditions. Basin of attraction. The logistic map.  The period doubling cascade. Sensitive dependence on initial conditions.
4 Itineraries and Sarkovskii's theorem. Chaos. Transition graphs and symbolic dynamics.
5 Fractals. Cantor set, iterated function system, dynamics and fractals, fractional dimension.
4 Bifurcation of fixed points. Bifurcation diagrams.

Last revised August, 2006