Abstract
We consider second order ordinary differential equations describing periodically forced dynamical systems with one and a half degrees of freedom. First, we study Hill's equation where we investigate boundedness of solutions. We also construct a class of equations, including a class of Hill's equation, for which we can obtain closed form solutions. We continue to analyse the boundedness of solutions to nonlinear systems with sufficiently high and low frequency forcing, utilising averaging techniques and KAM theory. We then focus on the study of dissipative systems with parameters chosen in a region of parameter space for which the equilibrium points of the linearised systems are Lyapunov stable. In dissipative systems, many of the solutions which exist in the absence of dissipation are destroyed, leaving a finite set of attractive solutions. We investigate the basins of attraction of the attractive equilibrium points and periodic orbits. In particular we study how the basins of attraction change when the coefficient of dissipation is allowed to initially vary as a function of time. Although it is the final value of the dissipative coefficient which determines which attractors eventually exist, the sizes of their corresponding basins of attraction are found to depend strongly on the full evolution of the coefficient. We study the dynamics of systems with the dissipative parameter modeled by both linearly increasing and decreasing functions of time with various gradients. In this instance we outline four cases pertaining to the sets of attractors at both the initial and final values of the coefficient of dissipation. For each scenario we state our expectations which are illustrated by means of numerical simulation for the systems of the pendulum with vertically oscillating support and the pendulum with periodically varying length. Further to this, a method which allows the fast numerical computation of basins of attraction for a system with an initially varying coefficient of dissipation is identified. This method is also applied in explaining a phenomenon in which the basins of attraction can drastically change when the coefficient of dissipation is a function of time.