620-341 Dynamical Systems & Chaos

One of 620-113, 620-123, 620-143, [98]620-130, [98]620-132; and one of 620-221, 620-231, 620-232, 620-233, 620-234, 620-252.

This subject develops problem-solving skills and sharpens analytical skills. Students will work in groups, tackling unfamiliar problems. Each team will plan their project work and deliver both oral and written presentations.

This subject introduces the basic concepts and recent developments in the fields of dynamical systems and chaos, including stability of equilibria and renormalisation theory of transitions to chaos. Students should develop the ability to analyse simple nonlinear discrete and continuous dynamical systems, and to chart parameter regions of stability, periodicity and chaos. This subject demonstrates the power as well as the limitations of dynamical systems theory and chaos applied to realistic complex systems.

Dynamical systems topics include phase space, Poincare sections, phase portraits, invariant measures. Chaos topics include integrable and chaotic systems, maps on an interval, period doubling and universality, iteration in the complex plane, Mandelbrot and Julia sets, renormalisation and scaling, reversible mappings, KAM theorems, strange attractors, fractals, limit cycles, Hopf bifurcation, Lorentz attractor, Lyapunov exponents, dimensions of strange attractors, hierarchies of chaos; and applications to ecology, chemical reactions, economics, management and meteorology.

Up to 24 pages of written assignments and a 3-hour end-of-semester written examination.