Title:

Invariant manifolds of planar mappings

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 C^{1} 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 C^{1}. 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 C^{2}, 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.
