The advent of novel engineered or smart materials, whose properties can be
significantly altered in a controlled fashion by external stimuli, has
stimulated the design and fabrication of smaller, faster, and more
energy-efficient devices. As the need for even more powerful technologies
grows, networks have become popular alternatives to advance the fundamental
limits of performance of individual devices. Thus, in the first part of this
talk we provide an overview of fifteen years of work aimed at combining ideas
and methods from equivariant bifurcation theory to model, analyze and fabricate
novel technologies such as: ultra-sensitive magnetic and electric field
sensors; networks of nano oscillators; and multi-frequency converters.
In the second part of the talk, we discuss more recent work on networks of
vibratory gyroscopes systems. Under normal conditions of operation, the model
equations can be reformulated in a Hamiltonian structure and the corresponding
normal forms are then derived. Through a normal form analysis, we investigate
the effects of various coupling topologies and unravel the nature of the
bifurcations that lead a ring of gyroscopes of any size into and out of
synchronization. The synchronization state is particularly important because
it can lead to a significant reduction in phase drift, thus enhancing
performance. The Hamiltonian approach can, in principle, be readily extended
to other symmetry related systems.