Symmetry-Breaking Bifurcations and Patterns of Oscillations in Rings of Crystal Oscillators

Abstract

Symmetry is used to investigate the existence and stability of collective patterns of oscillations in rings of coupled crystal oscillators. We assume N identical crystal oscillators, where each oscillator is described by a two-mode nonlinear oscillatory circuit. We also assume the coupling to be identical and consider two different topologies, unidirectional and bidirectional, which lead to networks with Γ = ZN and Γ = DN symmetry, respectively. The whole system can be seen as an ε-perturbation of N uncoupled, two-mode oscillators. The spectrum of eigenvalues of the linearized system near the origin leads to expressions not amenable to analysis. To circumvent this problem, we apply the method of averaging and rewrite the model equations, via near identity transformations, in the socalled full-averaged equations. The truncation to the average part, expressed in complex coordinates, is O(2) × O(2) × Γ-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 for small ε. A decomposition of the phase-space dynamics along irreducible representations of the symmetry groups ZN and DN leads to a block diagonalization of the linearized averaged equations. Direct computation of eigenvalues 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. Numerical simulations are conducted to show representative examples of emergent rotating waves. The motivation for this work is to aid future design and fabrication of novel precision timing devices.

One-way dependent clusters and stability of cluster synchronization in directed networks

“…For instance, a large corpus of studies has shown the connection of neural synchronization of brain areas with cognition [6], animal locomotion is controlled to some extent by central pattern generators that impose a given gait by exploiting synchronous clusters of neurons [7,8,9], and connected vehicle systems often give rise to CS [10]. The study of CS is therefore relevant to analyze and control natural (ecological, immune, neural, and cellular) [11] and artificial [12,13,14]…”

Patterns of synchronized clusters in adaptive networks

“…In neural dynamics, synchronization of brain areas is believed to provide a cognition mechanism 7,8 , functional assemblies of brain neurons display distinct interdependent synchronous oscillations 9 , and central pattern generators exploit synchronous clusters of neurons to generate gaits in animal locomotion 10 . Therefore, the study of CS is relevant to analyzing and controlling both natural (ecological, social, immune, neural, and cellular) 11 and artificial 12 systems.…”