In chapter 3 we discussed the behaviour of a dynamical system when it is displaced from its state of rest, or equilibrium, and, in particular, we studied the conditions under which the displaced system does not wander too far from equilibrium or even converges back to it as time goes by. For such cases, we call the equilibrium stable or asymptotically stable. But what happens if we perturb an unstable equilibrium?

For an autonomous linear system, if we exclude unlikely borderline cases such as centres, the answer to this question is straightforward: orbits will diverge without bound.

The situation is much more complicated and interesting for nonlinear systems. First of all, in this case we cannot speak of the equilibrium, unless we have established its uniqueness. Secondly, for nonlinear systems, stability is not necessarily global and if perturbations take the system outside the basin of attraction of a locally stable equilibrium, it will not converge back to it. Thirdly, besides convergence to a point and divergence to infinity, the asymptotic behaviour of nonlinear systems includes a wealth of possibilities of various degrees of complexity.

As we mentioned in chapter 1, closed-form solutions of nonlinear dynamical systems are generally not available, and consequently, exact analytical results are, and will presumably remain, severely limited. If we want to study interesting dynamical problems described by nonlinear differential or difference equations, we must change our orientation and adapt our goals to the available means.