Aims & Learning Objectives:
Aims: The course is intended to be an elementary and accessible introduction to dynamical systems with examples of applications. Main emphasis will be on discrete-time systems which permits the concepts and results to be presented in a rigorous manner, within the framework of the second year core material. Discrete-time systems will be followed by an introductory treatment of continuous-time systems and differential equations. Numerical approximation of differential equations will link with the earlier material on discrete-time systems.
An appreciation of the behaviour, and its potential complexity, of general dynamical systems through a study of discrete-time systems (which require relatively modest analytical prerequisites) and computer experimentation.
Topics will be chosen from the following:
Discrete-time systems. Maps from 3n to 3n. Fixed points. Periodic orbits. α and ω limit sets. Local bifurcations and stability. The logistic map and chaos. Global properties.
Continuous-time systems. Periodic orbits and Poincaré maps. Numerical approximation of differential equations. Newton iteration as a dynamical system.