The investigator analyzes the large systems of coupled oscillators that arise in various branches of biology and condensed-matter physics. A common theme is the spontaneous emergence of synchronization due to cooperative interactions among the oscillators. The systems of ordinary differential equations that constitute the models will be investigated analytically and numerically. Techniques to be used include methods of dynamical systems, bifurcation theory, statistical visualization of the many-osillator dynamics plays a crucial heurisc role in the research. Several of the taksk are expected to generate experimentally testable predictions. The mathematical problems studied here are models of behavior as varied as the beating of a heart, the patterned flashing groups of fireflies, or the circadian rhythm. Specific areas to be studied include: phase models with a large but finite number of oscillators; pulse-coupled oscillators with distributed frequencies or local coupling; phase-locking and invariant 2-tori in arrays of Josephson junctions; arrays of electronic relaxation oscillators; coupled oscillator models for the Drosophila circadian system; and delayed switching of charge-density.