Unit information: Advanced Nonlinear Dynamics and Chaos in 2020/21

Unit name	Advanced Nonlinear Dynamics and Chaos
Unit code	EMATM0001
Credit points	10
Level of study	M/7
Teaching block(s)	Teaching Block 2 (weeks 13 - 24)
Unit director	Professor. Champneys
Open unit status	Not open
Pre-requisites	EMAT33100 Nonlinear Dynamics & Chaos
Co-requisites	None
School/department	School of Engineering Mathematics and Technology
Faculty	Faculty of Engineering

Description including Unit Aims

Students will be introduced to more advanced methods in nonlinear dynamics and shown further applications of this work to real systems. They will be taken to the frontiers of the subject, ready to deal with some of its most challenging problems. Material covered in this course will be selected from abstract definition of dynamical system, structural stability and topological definition of bifurcation, centre manifolds, normal forms, analysis of codimension-one and two bifurcations, periodic orbits and Poincare maps, quasi-periodic motion, resonance and parametric resonance, circle maps, KAM theory & Arnold tongues, synchronisation of oscillators, Smale horseshoes, symbolic dynamics, homoclinic bifurcations, 'Shilnikov-type' bifurcations, principles of numerical path-following method and bifurcation analysis, non-smooth bifurcations in piecewise continuous systems, fractals, Lyapunov exponents and chaotic attractors.

Aims:

1. The aim of the unit is to introduce students to advanced methods in nonlinear dynamics, covering maps, ordinary and partial differential equations. Students will also be shown applications of the theory to real-world systems. They will be taken to the frontiers of the subject, ready to deal with some of its most challenging problems.
2. The particular topics covered in the unit will be driven by current research. Potential topics include global bifurcation theory and computation, resonance and invariant tori, centre manifolds, KAM theory, Smale horseshoes, 'Shilnikov-type' bifurcations, principles of numerical path-following methods, bifurcations in piecewise smooth systems, ractals, Lyapunov exponents and chaotic attractors, quasi-periodic motion, sychronisation of coupled oscillators.

Intended Learning Outcomes

1. To have familiarity with a variety of research topics in applied nonlinear dynamics, covering both ordinary differential equations, and maps
2. To be equipped with a set of analytical tools for analysing bifurcations, chaos and other nonlinear effects in systems arising from applications

Teaching Information

Teaching will be delivered through a combination of synchronous and asynchronous sessions, including lectures, supported by live online sessions, problem sheets and self-directed exercises.

Assessment Information

1 Summative Assessment, 100% - Summer Timed Assessment. This will assess all ILOs.

Reading and References

• Elements of Applied Bifurcation Theory (2nd Ed)
• Yu. A. Kuznetsov, Springer, 1998