Since the flow near a periodic orbit can be described by a return map (which is as smooth as the original flow) the bifurcations of periodic orbits of differential equations and fixed points or periodic orbits of maps can be treated as one and the same topic. As we saw in Chapter 6, the linearized map near a fixed point of a nonlinear map is a good model of the behaviour near the fixed point provided the fixed point is hyperbolic, i.e. no eigenvalues of the linear map lie on the unit circle. The eigenvectors associated with eigenvalues inside the unit circle correspond to stable directions and those associated with eigenvalues outside the unit circle correspond to unstable directions. Furthermore, since hyperbolic fixed points of maps are persistent under small perturbations of the map (a result analogous to the persistence of hyperbolic stationary points in flows; cf. Chapter 4) there can be no local bifurcations near a hyperbolic fixed point so, as in the previous chapter, we are forced to consider nonhyperbolic fixed points. This is made easier by the centre manifold for maps, which plays the same role in this chapter as the Centre Manifold Theorem for flows did in the previous chapter.