- 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, transcritical, pitchfork.
- 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.
#### D. Chaos

- Lorenz equations.
- Homoclinic bifurcations.
- Symbolic dynamics. The Smale horseshoe.
- Cantor sets. Fractal dimension.

- 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, CUP 1990.
- R.L. Devaney, An introduction to chaotic dynamical systems, Addison-Wesley 1987.
- S. Wiggins, Introduction to applied nonlinear dynamical systems, Springer 1990.
- 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.).

The principal texts are