Dynamical Systems and Applications II (0580023) 20 credits

Aims: To continue the study of Dynamical Systems from Term I. The main topic is how to describe a chaotic system. There is no common definition of chaos which is widely accepted by the experts in the field. However, there are plenty of cases where everyone agrees there is an instance of chaotic behaviour. The module builds on this theme.

Learning objectives: The module should give the ability to know when to use deterministic nonlinear models to describe a real set of data and analyse its limitations and when to use it.

Syllabus: The main reference is the part called Chaos in the Strogatz's book in the recommended texts below. The module is accompanied by a practical part where the students are given specific examples to simulate on the computer.

  • Lorenz System and strange attractor: bifurcation diagram, Hopf bifurcation, transient chaos and strange attractor regions. The Lorenz map.
  • Iterations of maps: Poincaré section and return map. Attracting, repelling, neutral fixed points and periodic points.
  • The logistic family as a model of population dynamics: period-doubling bifurcations, Feigenbaum's universality. Lyapunov exponents.
  • Invariant Cantor sets and coding.
  • Existence of periodic orbits: Sarkovski's Theorem, Transition graphs.
  • Fractal dimension and fractals: box dimension, Hausdorff dimension, similarity dimension, pointwise and correlation dimension.
  • Iterated Function Systems. Fractal basin boundaries.
  • Strange attractors: baker's transformation. Smale's horseshoe (repeller).
  • Hénon map and Hénon attractor. Rössler system.
  • Stable and unstable manifolds. Homoclinic points and existence of horseshoes. Hyperbolic automorphisms of the torus.

    Recommended texts:

  • Nonlinear dynamics and chaos, S.H.Strogatz (S 7.38 STR).
  • Chaos and nonlinear dynamics: an introduction for scientists and engineers, by R.C.Hilborn (U 0.15 HIL).
  • Chaos: an introduction to dynamical systems, by K.Alligood, T.Sauer and J.Yorke (U 0.15 ALL).

    Teaching: Spring Term, 3 lectures per week.

    Assessment: Two hours closed examination in Easter Vacation. (90%), Coursework (10%).