Dynamical Systems 1

Description

Dynamical systems is a rapidly developing branch of Mathematics with growing applications in diverse fields from traditional areas of applied mathematics to numerical analysis, biological systems, economic models and medicine.

You may have heard of Lorenz’s famous question: “Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?”. The question was meant to portray one of the main concepts in chaos theory (sensitive dependence to initial conditions). This course will teach theory, techniques and applications of systems of nonlinear equations. In all cases, we are concerned with dynamics: the time evolution of spatial structure. We will cover the mathematics behind chaos theory, and learn techniques for analysing nonlinear systems. Since it is usually difficult or impossible to write down an exact solution to systems of nonlinear equations, the emphasis will be on qualitative techniques for classifying and understanding the behaviour of nonlinear systems.
Both main types of dynamical system will be studied: discrete systems, consisting of an iterated map; and continuous systems, arising from nonlinear differential equations. The natural relationships between discrete and continuous time systems will be emphasised too. Probabilistic properties may be mentioned — chaotic systems exhibit a certain amount of randomness — but the main focus will be on topological and smooth dynamics. (One must at least be able to define continuity, and differentiability helps!)

This course is independent of Math363 Dynamical systems, although previous enrolment there is desirable.

Topics covered:

• Maps and flows as dynamical systems (including return maps).
• Chaotic behaviour of one-dimensional maps, including period-doubling.
• Symbolic dynamics.
• Hyperbolicity, linearisation and stability of fixed points and period orbits.
• Invariant manifolds and phase portrait analysis.
• Centre manifolds and local bifurcations; global bifurcations.
• Topic(s) selected from: fast-slow systems; Lyapunov exponents, entropy and randomness; non-autonomous dynamics.