Introduction to the theory of stability

In the chapters three and four we have seen equilibrium solutions and periodic solutions. These are solutions which exist for all time. In applications one is often interested also in the question whether solutions which at t = t₀ are starting in a neighbourhood of such a special solution, will stay in this neighbourhood for t > t₀. If this is the case, the special solution is called stable and one expects that this solution can be realised in the practice of the field of application: a small perturbation does not cause the solutions to move away from this special solution. In mathematics these ideas pose difficult questions. In defining stability, this concept turns out to have many aspects. Also there is of course the problem that in investigating the stability of a special solution, one has to characterise the behaviour of a set of solutions. One solution is often difficult enough.