In this chapter we study the behaviour of the solutions of IVP for t ? ?. In 1 the various possibilities for the long term behaviour of solutions are mentioned. One possibility is chaotic behaviour. To this phenomenon a separate chapter (VI) is devoted. In the present chapter we focus on convergence to stationary points and periodic solutions. In 2 we point out that there are many notions of stability and give the most common definitions. Linear systems play a special r le in stability analysis, because for those systems explicit results can often be found. This is worked out for systems with constant coefficients in 3. For nonlinear systems one can rely on two different approaches: linearisation and Lyapunov functions. The technique of linearisation is presented in 4. It uses the fact that most nonlinear vector fields can be approximated by a linear one in the vicinity of a stationary or periodic solution. Lyapunov functions are introduced in 5. Both linearisation and Lyapunov functions yield local information. A global analysis of the stability properties of nonlinear systems is difficult in general. For planar systems one can investigate the problems in more detail, as is illustrated in 6. The important case of periodic systems is considered in 7. Finally, we deal with the stability of ?-equations in 8.

1. Introduction

In this section we study the effect of perturbations of the IVP

on its solution x. Perturbations may concern both the initial value, the initial time, and the...