Patterns at primary Hopf bifurcations at a plexus of identical oscillators

Mathematically, an oscillator is a system of ordinary differential equations with a periodic limit cycle, usually arising from a Hopf bifurcation for a stationary solution. A plexus is a collection of such oscillators, coupled to each other via one-way or two-way communication channels. Plexuses model a wide number of phenomena in the mathematically oriented sciences. The primary Hopf bifurcation theory of such a plexus is developed. There are features of the bifurcating solutions which depend only on the combinatorics of the plexus and not on the details of the individual oscillators. Such features are called the pattern of the bifurcation. The relation of patterns to the modes of small oscillations of mechanical systems is discussed. More detailed information is developed for weak coupling. The theory is used to analyze two particular types of plexuses; viz, rings and lines of oscillators. It is then shown that the structure of a plexus permits numerical bifurcation computations to be made more efficien...