Noise-Assisted Synchronization in Networks of Coupled Mathieu-Duffing Oscillators

Abstract Studies of collective behavior of coupled dynamical entities continue to reveal a broad range of phenomena, including synchronization. This behavior has often been analyzed by using models such as the Kuramoto model. The dynamics of this model has been related to dynamics of a diverse range of systems, for example, Josephson junctions, laser systems, power grids, and even, systems in sociology. With mechanical systems, synchronization has been commonly analyzed in systems, which exhibit limit cycle oscillations. Utilizing limit cycle oscillations allows one to draw analogies with the Kuramoto model. Here, the authors analyze a network consisting of parametrically excited nonlinear oscillators and demonstrate that by introducing random interventions, one can promote synchronous collective behavior. Specifically, in the uncoupled limit, the method of averaging is utilized to show that a parametric excitation results in multiple stable steady states. For moderate coupling strengths, this phase space geometry remains largely intact and there exists a multitude of stable steady states. Most of these solutions correspond to asynchronous behavior. In addition, the emergence of cluster synchronization, where multiple clusters of adjacent oscillators with equal phase coexist, is observed. When a weak stochastic input is introduced, the asynchronous behavior is suppressed, and after an initial transient phase, collective behavior is found to emerge. Thus, adding temporal disorder (i.e., noise) to disordered dynamics can generate order (e.g., synchronization). This counterintuitive effect suggests applications in systems ranging from rotating machinery to Josephson junction arrays.