In recent years there has been unprecedented popular interest in the chaotic behaviour of discrete dynamical systems. The ease with which a modest microcomputer can produce graphics ofextraordinary complexity has fired the interest of mathematically-minded people from pupils in schools to postgraduate students. At undergraduate level, there is a need to give a basic account of the computed complexity within a recognized framework of mathematical theory. In producing this replacement for Ordinary Differential Equations (ODE) we have responded to this need by extending our treatment of the qualitative behaviour of differential equations. This book is aimed at second and third year undergraduate students who have completed first courses in Calculus of Several Variables and Linear Algebra. Our approach is to use examples to illustrate the significance of the results presented. The text is supported by a mix of manageable and challenging exercises that give readers the opportunity to both consolidate and develop the ideas they encounter. As in ODE, we wish to highlight the significance of important theorems, to show how they are used and to stimulate interest in a deeper understanding of them.