Chaotic Dynamical Systems
Listed as Applied Mathematics Topic D in the Course Planner.
Go to this course in the University Course Planner.
Many complex physical systems can be modelled accurately by a small number of deterministic
coupled differential or difference equations.
Examples include neuron dynamics, population modelling, chemical reactions, stirring and mixing,
particle interactions, forced pendulums, weather modelling, and even dripping taps, to name just a
few. Though these models are often simple to write down, their solutions often exhibit very
complicated behaviours (which are observed in reality), including chaos, meaning extreme sensitivity
to initial conditions.
This course will provide an introduction to the theory of continuous and discrete dynamical systems, with a particular emphasis on bifurcations and routes to chaotic behaviour. The first few lectures will set the scene, showing how low-dimensional dynamical systems arise from various approximations to complex physical systems. We will observe that these systems can exhibit exotic changes in behaviour as governing parameters are varied smoothly, including chaotic dynamics. To understand this behaviour we will study in detail the stability and bifurcations of periodic structures in these systems, bringing together results from linear algebra, multivariable calculus, differential equations, topology and group theory.
By the end of the course, students should be able to understand/formulate simple mathematical models of physical phenomena, know how to reduce these models to either discrete or continuous dynamical systems, and be able to extract some of the basic properties of the solutions of such systems. They should also have an appreciation of precisely what chaos is,
when it can occur, and how to quantify it.
Topics covered will include: an introduction to flows and maps; phase plane analysis of autonomous flows; periodic orbits and limit cycles; the Poincare-Bendixson theorem; saddle-node, transcritical, pitchfork and Hopf bifurcations; conservative versus dissipative systems; hysteresis; dimensional reduction and Poincare sections; analysis of interval maps; period doubling bifurcations and cascades; chaos; Sharkovskii's theorem; area preserving maps and homoclinic tangles; the Thurston-Nielsen theorem; mapping classes and braid groups (time permitting).
This course is not recorded as prequisite for other courses.
Differential equations, dynamical systems & an introduction to chaos by
Hirsch, Smale and Devaney. Published by Elsevier.