The treatment of bifurcations in previous chapters has been based on a local analysis of the flow near a stationary point or a periodic orbit. There is another class of bifurcations which is just as fundamental as these local bifurcations but which depends on global properties of the flow. A simple example of such a bifurcation is shown in Figure 12.1. We consider a planar flow depending upon a parameter µ such that for µ > 0 there is a stable periodic orbit enclosing an unstable focus and a saddle outside the periodic orbit with stable and unstable manifolds as sketched in Figure 12.1a. As µ decreases to zero the periodic orbit approaches the saddle and when µ = 0 the periodic orbit collides with the saddle to form an orbit called a homoclinic orbit. So at µ = 0 one branch of the stable manifold of the saddle coincides with a branch of the unstable manifold of the saddle. By Peixoto's Theorem (Theorem 4.13) this is structurally unstable, and so typical small perturbations of the system will not have a homoclinic orbit, and in µ < 0 (Fig. 12.1c) this connection has been broken and we see that the periodic orbit has disappeared. Hence the homoclinic bifurcation at µ = 0 has destroyed the periodic orbit. We shall see that more complicated homoclinic bifurcations can create chaotic behaviour, but to begin with we shall stick to the two-dimensional situation, starting with an example.