As the theory of dynamical systems and chaos develops, more and more recent results are filtering through to undergraduate courses. The aim of this book is to provide a coherent account of the qualitative theory of ordinary differential equations which deals in an even handed and consistent way with both the ‘classical’ results of Poincaré and Liapounov and the more recent advances in bifurcation theory and chaos. The book covers two undergraduate courses: a first course in nonlinear differential equations and an introduction to bifurcation theory.

Throughout, the emphasis is on understanding and the ability to apply theory to examples rather than on rigorous mathematical developments. Although there are theorems, the level of rigour is not that of a pure mathematical text. None the less, it is vital to appreciate the restrictions and limitations of any method and so wherever possible I have stated results in a precise form. The choice of topics has also been influenced by a desire to cover material which can be examined sensibly.

This book has developed out of courses given to third year undergraduates at the University of Warwick and the University of Cambridge. In both places I have been fortunate to inherit the notes of previous lecturers: Tony Pritchard at Warwick, Peter Swinnerton-Dyer, John Hinch and others at Cambridge. The first seven chapters owe an enormous debt to these people and in some places it is hard for me to see where they end and I begin (although I retain full responsibility for any errors).