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For 2013-2014 I shall be teaching a course in nonlinear systems (MATHGM02) in Mathematics and a Ship Dynamics course as part of the MSc in Naval Architecture. The syllabus for GM02 is below.
Internal UCL students who are registered on the course can visit the UCL Moodle page
GM02 Nonlinear Systems Syllabus
This component aims to give an overview of the main aspects of nonlinear systems and to provide definitions and theoretical background
1. Continuous Dynamical Systems:
Equilibria. Local and global stability. Liouville's Theorem. Conservative and dissipative mechanical systems. Bifurcation analysis for one- and two-dimensional systems, including Hopf bifurcation. Periodic solutions and Poincare-Bendixson theorem. Perturbation methods. Bifurcation theory for one- and two-dimensional systems, including Hopf bifurcation. Applications: non-linear oscillators, Hamiltonian systems, dissipative systems.
2. Discrete Dynamical systems:
Iterated maps as dynamical systems in discrete time. The logistic map as main example. Equilibria, cycles and their stability. Period doubling bifurcations. Simple random properties of discrete trajectories. Elementary properties of maps in two dimensions. Lyapunov exponents, attractors and the butterfly effect.
3. Non-linear waves:
Linear waves, dispersion relations, dispersion versus dissipation, stable and unstable waves. Travelling wave solutions of non-linear partial differential equations, for example the Korteweg-de Vries, non-linear Schrodinger equations. Phase-plane analysis, solitons
Recommended Books :
S.H.Strogatz, Nonlinear Dynamics and Chaos, Perseus Books, 1994.
J.M.T.Thompson & H.B. Stewart, Nonlinear Dynamics and Chaos, Wiley 2003.
E.Ott, Chaos in Dynamical Systems, CUP, 1993.
D.K.Arrowsmith & C.M.Place, Dynamical Systems, Chapman Hall
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer
L.D.Landau & E.M.Lifshitz, Course of Theoretical Physics, Vol. 1 Mechanics, Pergamon
Drazin and Johnson, Solitons, Cambridge Texts
and other books by Drazin, e.g. Nonlinear Systems
Assessed via a 2 hour examination.
You can download the Syllabus here