## 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.

#### Prerequisites

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

### Synopsis

#### 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
• 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, 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.).