Abstract
This thesis involves the analysis of four classes of nonlinear oscillators. We investigate a damped planar pendulum subject to vertical sinusoidal displacement of appropriate amplitude and frequency, a Hamiltonian planar pendulum with support point oscillating in the vertical direction, a forced spherical pendulum as a constrained dynamical system and a spinning double pendulum with the two masses oscillating in transversal planes. The motivation for this research was to understand and determine the fundamental dynamical properties of the four model systems. For this purpose analytical and numerical tools have been employed. Linearization, phase portraits, Poincare sections, basins of attraction, KAM theory, Lyapunov exponents and normal form theory have been considered as examples. For the damped planar pendulum a rigorous analysis is presented in order to show that, in the presence of friction, the upward equilibrium position becomes asymptotically stable. Furthermore, using numerical tools, the dynamics of the system far from its equilibrium points is systematically investigated. For the undamped and parametrically perturbed planar pendulum, we use KAM type arguments to rigorously prove the stability of the equilibrium point corresponding to the upside-down position. For the spherical pendulum a numerical framework is developed, which allows orbits to explore the entire sphere. We show that the qualitative change in the Poincare sections from regular to chaotic behaviour is in excellent qualitative agreement with corresponding computations of the Lyapunov exponents. Finally we study the dynamics of the spinning double pendulum by using normal form theory. We have identified the regions in physical parameter space where a codimension-two singularity occurs. An algorithm for the Cushman-Sanders normal form is constructed and analyzed. A representative model for the truncated normal form is presented.