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
- Phase plane analysis: saddle, node, focus, centre.
- Bifurcations for 2-D systems: saddle-node, transcritical, pitchfork,
Hopf. The Hopf bifurcation theorem.
- Normal forms: construction. Resonance. Poincaré-Dulac theorem.
- Normal forms: examples. Embedding at bifurcations.
B. Bifurcation theory for maps
- Poincaré maps.
- Bifurcation for 1-D maps: saddle-node,
- Period-doubling; chaos.
- Symbolic dynamics; the tent map.
C. Perturbation theory
- Poincaré-Lindstedt method.
- Method of averaging.
- Resonance and stochasticity.
- Poincaré-Birkhoff theorem. Homoclinic
points. Duffing equation.
- Lorenz equations.
- Homoclinic bifurcations.
- Symbolic dynamics. The Smale horseshoe.
- Cantor sets. Fractal
These can be downloaded as postscript or pdf files; currently available are
- Problem sheets ps or pdf.
- Finals questions
- Specimen/revision questions (under revision):
ps or pdf
The principal texts are
- P. Drazin, Nonlinear systems, CUP 1992.
- P. Glendinning, Stability, instability and chaos, CUP 1994.
- I. C. Percival and D. Richards, Introduction to dynamics, CUP 1985.
Subsidiary texts are
- D. K. Arrowsmith and C.M. Place, An introduction to dynamical systems,
- R.L. Devaney, An introduction to chaotic dynamical systems,
- S. Wiggins, Introduction to applied nonlinear dynamical systems,
- M. Tabor, Chaos and integrability in nonlinear dynamics: an
introduction, Wiley 1989.
- J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical
systems, and bifurcations of vector fields, Springer 1983.
- D. W. Jordan and P. Smith, Nonlinear ordinary differential equations, OUP
1987 (2nd ed.).