| Abstract: |
| Symmetry is used to investigate the existence and stability of collective patterns of oscillations in rings of identically coupled crystal oscillators. We assume $N$ identical crystal oscillators, where each oscillator is described by a two-mode nonlinear oscillatory circuit. We consider two different topologies, unidirectional and bidirectional, which lead to networks with $\Gamma= Z_N$ and $\Gamma=D_N$ symmetry, respectively. We apply the method of averaging and show that the full-averaged system is $\Gamma \times O(2) \times O(2)$-equivariant. Then, we present new theoretical results linking symmetry and averaging theory to study the existence and stability of steady-states of the truncated averaged systems, and show that they persist as periodic solutions of the full-averaged system. Computation of eigenvalues via the isotypic decomposition leads to the desired identification of periodic solutions that emerge via symmetry-preserving and symmetry-breaking steady-state bifurcations leading to the corresponding periodic solutions with spatio-temporal symmetries. Then, we provide analytical proof of the scaling laws of precision timing devices, based on symmetric networks, and show that $1/N$ is the fundamental limit of phase-error reduction that one can obtain with a symmetric network of nonlinear oscillators of any type, not just crystals. |
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