3509 (Dynamical Systems)
Old Code: MATH3509
Value: Half unit (= 7.5 ECTS credits)
Structure: 3 hour lectures per week. Compulsory assessed coursework.
Assessment: 90% examination, 10% coursework
Normal Pre-requisites: Basic linear algebra and methods course material taken in years 1
and 2 of a mathematics degree.
Lecturer: Dr M Banaji
Course Description and Objectives
The idea of a dynamical system, with examples, and the notions of solutions, orbits, limit
sets and stability. Graphical analysis of 1D maps. Local stability of ﬁxed and periodic points
of maps. The doubling map of the circle used to illustrate dense periodic points, topological
transitivity and sensitive dependence, leading to deﬁnitions of chaos. Proof that a period 3
orbit for a 1D map implies chaos. Rewriting ODEs as ﬁrst order systems. Vector ﬁelds and
isoclines. Local stability of ﬁxed points and periodic orbits of ODEs. Global stability and
Lyapunov functions. Conservative and gradient systems. The Poincare-Bendixson theorem and
a full classiﬁcation of limit sets in two dimensions. An introduction to bifurcation theory.
Robert Devaney, An introduction to chaotic dynamical systems.
Paul Glendinning, Stability, instability, and chaos: an introduction to the theory of nonlinear
D K Arrowsmith and C M Place, An introduction to dynamical systems.
Graphical analysis of 1D maps. The structure of limit sets in the quadratic family for diﬀerent
parameter values. Notion of prime period of a periodic orbit, and techniques to ﬁnd points of
prime period n > 1. The doubling map on the circle used to illustrate topological transitivity,
sensitive dependence, and Lyapunov exponents. Basic symbolic dynamics for 1D maps. Basic
ideas to do with higher dimensional systems.
Rewriting of higher order ODEs as ﬁrst order systems. Autonomous and non-autonomous
ODEs. Vector ﬁelds, isoclines. Sketching solutions based on vector ﬁelds. Complete classiﬁ-
cation of behaviour of linear ODEs. Linearisation of nonlinear ODEs about ﬁxed points and
periodic orbits. Qualitative techniques for nonlinear ODEs. Lyapunov functions.
Introduction to bifurcation theory Co-dimension 1 bifurcations of maps and vector ﬁelds.
How to recognise them, and classify them. Eigenvalues of linearised systems and recognising
various bifurcation scenarios.
May 2008 3509