Dynamics and bifurcations in a DnDn-symmetric Hamiltonian network. Application to coupled gyroscopes

• Rings of forced symmetrically coupled Hamiltonian gyroscopes are studied.
• The bidirectional network architecture preserves the Hamiltonian character.
• Linear analysis provides a threshold for synchrony for all ring sizes.
• Bifurcation is studied via Birkhoff normal form and the equivariant splitting lemma.

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 devices grows, networks have become popular alternatives to advance the fundamental limits of performance of individual units. In many cases, the collective rhythmic behavior of a network can be studied through the classical theory of nonlinear oscillators or through the more recent development of the coupled cell formalism. However, the current theory does not account yet for networks in which cells, or individual units, possess a Hamiltonian structure. One such example is a ring array of vibratory gyroscopes, where certain network topologies favor stable synchronized oscillations. Previous perturbation-based studies have shown that synchronized oscillations may, in principle, increase performance by reducing phase drift. The governing equations for larger array sizes are, however, not amenable to similar analysis. To circumvent this problem, the model equations are now reformulated in a Hamiltonian structure and the corresponding normal forms are derived. Through a normal form analysis, we investigate the effects of various coupling schemes and unravel the nature of the bifurcations that lead a ring of gyroscopes of any size into and out of synchronization. The Hamiltonian approach can, in principle, be readily extended to other symmetry-related systems.