This thesis deals firstly with work performed on the dynamics of models of three modulated systems. The concept of periodicity and its effect on scattering response functions from a harmonic lattice is initially examined, to demonstrate the variety of results that occur for differing periodic configurations. Contrasting our examples, we define an extra constraint on the periodicity of the system, translational invariance, the presence of which changes the spectrum of observables from being that of isolated delta functions, to an extended response. This distinction provides one of the main topics of interest. We give examples of how this extra condition manifests itself in the results for three distinct models. We examine the density of states for two dimensional electrons in a perpendicular magnetic field, the dynamic susceptibility of spin phases in magnetic models, and the modulated spring model, which describes the dynamics of crystal phases. The algebraic approach used is another area of interest, and relies on the periodicity characteristic to the system, being rational, Q= 2πM/N. This allows us to form equations of motion for the systems, characterised by three term recursion relations with periodic coefficients, which for rational Q have closed solutions. The analytic, computational, and conceptual investigation of the effect of taking increasingly accurate approximations to an irrational Q, (M/N→c, as N→∞) forms the bulk of this work. Secondly in a separate piece of work we consider the relation between an XY magnetic model with a random b z magnetic field, to Bose condensation in a random potential. Using a semi-classical quantisation, we find the low energy dynamics are governed by a ground state formed by strongly coupled spin pairs. We examine the effect of disorder and number of spin states on the stability of this ground state, and finally map this system to that of Bose condensation, where the role of the spin states is taken by particle interaction.