This book is an introduction to the theories of bifurcation and chaos. It treats the solution of nonlinear equations, especially difference and ordinary differential equations, as a parameter varies. This is a fascinating subject of great power and depth, which reveals many surprises. It requires the use of diverse parts of mathematics – analytic, geometrical, numerical and probabilistic ideas – as well as computation. It covers fashionable topics such as symmetry breaking, singularity theory (which used to be commonly called catastrophe theory), pattern selection, chaos, predictability, fractals and Mandelbrot sets. But it is more than a fashionable subject, because it is a fundamental part of the theory of difference and differential equations and so destined to endure. Also the theory of nonlinear systems is applied to diverse and countless problems in all the natural and social sciences, and touches on some problems of philosophy.

The writing of the book evolved with lecture courses I have given to final-year undergraduates at the University of Bristol and to graduates at the University of Washington and Florida State University in the USA over the last decade. I hope that others will enjoy this book as our students have enjoyed the courses.

Most of the equations treated in traditional mathematics courses at university are linear. These linear algebraic, ordinary differential, partial differential and integral equations are solved by various powerful methods, which essentially depend upon the principle of superposition.