The thesis is divided into two parts, the first dealing wtih closed invariant curves of dissipative mappings of the cylinder, and the second dealing with the splitting of separatrices for a particular area preserving cylinder map. Chapters 1 and 2 provide some motivation for the main results of Part I. Chapter 1 contains a brief discussion of Hopf bifurcation theory, and also an analysis of a dissipative 'standard map' of the cylinder, while Chapter 2 surveys some of the literature on the breakup of invariant curves for dissipative systems, with particular emphasis on the renormalisation approach to the subject. The main results of Part I are contained in Chapter 3, and depend on the concept of normal hyperbolicity of invariant manifolds. This condition guarantees that an invariant curve of a C^r map f is itself C^r, and that it persists under small perturbations to f. We prove that a C¹ invariant curve Γ of f is normally hyperbolic if the rotation number of f restricted to Γ is irrational. One consequence of this result is that if an irrational invariant curve is at the point of breakup, as described in Chapter 2, then it cannot be C¹. Part II of the thesis is concerned with the splitting of separatrices for the rapidly forced pendulum system θ = φ φ = sin θ + δcos t/ε for δ> 0 fixed and ε→ 0. In Chapter 4 we explain why the classical methods of analysis do not work, and introduce the PoincaréMap f of the system. This is a map of the cylinder, which for ε small enough, has a hyperbolic fixed point. We prove that the associated stable and unstable manifolds intersect. In Chapter 5 we extend f into C², and construct parametrisations γ-(τ) and γ+ (τ) for the stable and unstable manifolds respectively, such that γ±(τ+ 1) = f(γ±(τ)). We find that for Rτlarge and negative, γ+(τ) is close to γ(τ) = (4 tan^-1 e^2πepsilonτ, 2sech 2πepsilonτ), the parametrisation of the separatrix for the case δ = 0. This is also true for γ-(τ) when Rτ is large and positive.