Patterns of patterns of synchronization: Noise induced attractor switching in rings of coupled nonlinear oscillators

Abstract

Following the long-lived qualitative-dynamics tradition of explaining behavior in complex systems via the architecture of their attractors and basins, we investigate the patterns of switching between qualitatively distinct trajectories in a network of synchronized oscillators. Our system, consisting of nonlinear amplitude-phase oscillators arranged in a ring topology with reactive nearest neighbor coupling, is simple and connects directly to experimental realizations. We seek to understand how the multiple stable synchronized states connect to each other in state space by applying Gaussian white noise to each of the oscillators' phases. To do this, we first identify a set of locally stable limit cycles at any given coupling strength. For each of these attracting states, we analyze the effect of weak noise via the covariance matrix of deviations around those attractors. We then explore the noise-induced attractor switching behavior via numerical investigations. For a ring of three oscillators we find that an attractor-switching event is always accompanied by the crossing of two adjacent oscillators' phases. For larger numbers of oscillators we find that the distribution of times required to stochastically leave a given state falls off exponentially, and we build an attractor switching network out of the destination states as a coarse-grained description of the high-dimensional attractor-basin architecture.

Fast and slow domino regimes in transient network dynamics

“…This includes work on stochastic resonance on networks [21,34]. For example, [12] uses a master equation approach while [2,17] consider noise-induced switching of bistable nodes in complex networks. Much of this work aims to explain properties of attracting (statistically steady) states perturbed by noise; nonetheless, many important questions are related to the transient dynamics of networks affected by noise.…”

Complex Dynamical Networks Constructed with Fully Controllable Nonlinear Nanomechanical Oscillators

“…Examples of intriguing experimental avenues opened by this technology are found in our companion study involving analytical and numerical simulations. 29 It is noteworthy that the electronic components we employ to construct our modular nodes are fully integratable on-chip via application specific integrated circuit (ASIC) technology. Accordingly, the platform we describe here, constructed from discrete components, serves as a prototype for a next generation of massively-upscaled analog synchronized networks, which can be patterned by co-integration of 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 NEMS and CMOS VLSI.…”

Introduction to focus issue: Patterns of network synchronization

“… D'Huys et al [80] examine networks of Boolean switches with time-delayed coupling both experimentally and analytically, finding surprisingly long transients that scale exponentially with the time delay.  Emenheiser et al [81] investigate the deterministic and stochastic dynamics of nearest-neighbor-coupled micromechanical resonators, and characterize the switching that can occur between distinct attracting states.…”