Motivated by numerical studies of a peculiar ring of Van der Pol oscillators presented at the 2009 International Symposium on Nonlinear Theory and its Applications, Sapporo, Japan, a rigorous analysis of the collective behavior of the ring is carried out in this thesis work. The ring is peculiar because the individual oscillators are ?almost-identical? with only small differences in operation frequencies. Numerical simulations reveal collective patterns of behavior that include: complete synchronization regimes in which all oscillators oscillate with the same amplitude and same phase while they lock into a common frequency, despite the differences in their individual frequencies; and out of phase patterns where certain oscillators are out of phase with respect to others. Two rigorous approaches are combined in order to explain the existence and stability of these patterns and transitions between them: perturbation methods and analysis of amplitude-phase equations in normal form. Together these techniques provide the necessary tools for reducing the dimensionality of the governing equations and for generating bifurcation diagrams and analytical approximations to the bifurcation curves that separate different patterns of behavior.
