Nonlinear systems

Aims and objectives

The course aims to provide an introduction to the tools of dynamical systems theory which are essential in the study of many branches in the natural sciences. Areas of application include nonlinear instabilities and transitions in fluids, mathematical biology, celestial mechanics, amongst many others.


b5 is very useful, and those who have not done this will need to do some reading on phase plane analysis.


    A. Bifurcation theory for o.d.e.s

  1. Phase plane analysis: saddle, node, focus, centre.
  2. Bifurcations for 2-D systems: saddle-node, transcritical, pitchfork, Hopf. The Hopf bifurcation theorem.
  3. Normal forms: construction. Resonance. Poincaré-Dulac theorem.
  4. Normal forms: examples. Embedding at bifurcations.

    B. Bifurcation theory for maps

  5. Poincaré maps.
  6. Bifurcation for 1-D maps: saddle-node, transcritical, pitchfork.
  7. Period-doubling; chaos.
  8. Symbolic dynamics; the tent map.

    C. Perturbation theory

  9. Poincaré-Lindstedt method.
  10. Method of averaging.
  11. Resonance and stochasticity.
  12. Poincaré-Birkhoff theorem. Homoclinic points. Duffing equation.

    D. Chaos

  13. Lorenz equations.
  14. Homoclinic bifurcations.
  15. Symbolic dynamics. The Smale horseshoe.
  16. Cantor sets. Fractal dimension.

Course materials

These can be downloaded as postscript or pdf files; currently available are