In 1967 Winfree proposed a mean-field model for the spontaneous synchronization of chorusing crickets, flashing fireflies, circadian pacemaker cells, or other populations of biological oscillators [Win67]. The model consists of a large number of coupled nonlinear oscillators with randomly distributed frequencies, and is known to exhibit the temporal analog of a phase transition: for small coupling, the system behaves incoherently, with each oscillator running close to its natural frequency; but when the coupling exceeds a critical value, part of the population spontaneously synchronizes to a common frequency, resulting in collective oscillations.; In this thesis we present the first bifurcation analysis of the model, for a tractable special case. Numerical simulations reveal that the system displays rich collective dynamics as a function of two parameters; the coupling strength and the width of the frequency distribution. Besides incoherence, frequency locking, and oscillator death (a total cessation of motion caused by excessively strong coupling), there exist various hybrid solutions that combine two or more of these states. We provide a summary of the long-term collective behavior of the system in the form of a stability diagram, outlining the regions in parameter space where each state occurs. By analyzing the model in the limit of infinite system size, we are able to obtain exact formulas for three states (death, partial death, and incoherence), as well as their associated stability boundaries. |