Non-linear Dynamical Systems

COURSE OBJECTIVE

This course is meant to be a bridge from an introductory (undergraduate) to a specialized (postgraduate) course in dynamic systems. Our main focus is to give a taster of modern research topics in dynamical systems, and expose analytical and numerical methods, as well as applications. We present 2 separate subtopics in dynamical systems, giving an overview of different approaches and methods.

After finishing the course, the student:

• has learned about a selection of intermediate and advanced topics in dynamical systems
• knows the fundamentals of Newton mechanics, Lagrange mechanics, and Hamiltonian mechanics of particles
• knows how to formulate a problem in classical mechanics as a variational problem in infinite dimensions
• understands the concept of energy and of symmetry in classical physics
• has seen the extension to classical field theory and to quantum theory
• can write down conditions for the existence of steady states in nonlinear PDEs
• can investigate the linear stability of steady states using analytical and numerical methods
• can predict the occurrence of spatio-temporal patterns emerging from instabilities
• can classify bifurcations of steady states in nonlinear PDEs
• can produce numerical evidence in support of analytical predictions

COURSE CONTENT

The course is split into 2 independent parts:

1. Lagrangian and Hamiltonian dynamics in physics: Classical physics roughly splits into two parts: classical mechanics of particles and classical field theory. The rigorous mathematical formulation of the underlying dynamics led to the Lagrangian as well as the closely related Hamiltonian formalism. Both are based on the least action principle: the solutions of the equations of motion are critical points (minimizers) of a corresponding nonlinear action function(al).
2. Dynamics for spatially-extended systems: This section presents an introduction to pattern formation in spatially-extended systems (PDEs or integro-differential equations). We will look at how bifurcation theory for ODEs can be extended to PDEs and other infinite-dimensional dynamical systems. We will also see how bifurcations and instabilities give rise to solutions that contains spatial and temporal patterns, which are often observed in real-world phenomena. This part will involve (formal) analytical calculations, as well as numerical computations.

TEACHING METHODS 

One lecture and one exercise class per week.

METHOD OS ASSESSMENT 

Two written (take home/on campus) exams with weights 50%+50%. The grades of these assessments are cancelled in case the student takes a resit.