Entry requirements

120 credits including 90 credits in mathematics, or 60 credits in mathematics and 30 credits in scientific computing including Scientific Computing II and Scientific Computing III.

Learning outcomes

This course aims to provide insight and practice in how dynamical system models (i.e. partial differential equations, differential equation and difference equation) can be used to better understand scientific problems. The focus will be on model analysis, based both in theoretical analysis and numerical simulations.

In order to pass the course the student should be able to

• outline the mathematical methods and techniques used to analyse these models and understand in what situations these methods can be applied;
• understand how to draw a conclusion from a model;
• use a computer package to investigate models numerically;
• numerically computate invariant manifolds (i.e. fixed points, attached invariant manifolds...) and understand their role in the composition of the phase portrait:
• numerically compute dynamical observables (i.e. Lyapunov exponents, Hausdorff dimensions...) and understand their role.

Content

The course will be driven by a series of cases studies. In each case study there will be an emphasis on both the mathematical analysis (i.e. techniques), and numerical simulations.

Some case studies that could be covered in the course are:

Molecular and cellular biology. Non-dimensionalisation; Michaelis-Menten kinetics; matched asymptotic expansions; models of neural firing; oscillations in biochemical systems.

Coupled oscillators. Flows on the circle; driven and coupled pendulums; global bifurcations; firefly flashing; Kuramoto model.

Biological motion. Introduction to partial differential equations; diffusion equation; Fisher's equation; travelling wave solutions; reaction-diffusion equations and pattern formation; Turing bifurcations.

Chaos. Population dynamics and one-dimensional maps. Cobweb diagrams; periodic windows; Liapunov exponent.

The n-body problem.

Instruction

Lectures and problem solving sessions.

Assessment

The course contents will be assessed by means of hand-in exercices comprising both analytical and numerical aspects.