Exotic states in a simple network of nanoelectromechanical oscillators

Abstract

Synchronization of oscillators, a phenomenon found in a wide variety of natural and engineered systems, is typically understood through a reduction to a first-order phase model with simplified dynamics. Here, by exploiting the precision and flexibility of nanoelectromechanical systems, we examined the dynamics of a ring of quasi-sinusoidal oscillators at and beyond first order. Beyond first order, we found exotic states of synchronization with highly complex dynamics, including weak chimeras, decoupled states, traveling waves, and inhomogeneous synchronized states. Through theory and experiment, we show that these exotic states rely on complex interactions emerging out of networks with simple linear nearest-neighbor coupling. This work provides insight into the dynamical richness of complex systems with weak nonlinearities and local interactions.

Phase reduction beyond the first order: The case of the mean-field complex Ginzburg-Landau equation

“…In summary, in this work we achieve second-and thirdorder phase reductions of the MF-CGLE. In our view, higherorder phase reductions promise to be crucial for our understanding of collective chaos and other exotic phenomena [12]. Moreover, analytic higher-order phase reductions may also serve as test beds for numerical phase reductions recently implemented [63].…”

“…which can be inserted in (12) to obtain the ǫ 2 contribution. Through elementary manipulations the second-order phase reduction of Eq.…”

“…This procedure was followed in [12,53], with the difference being that there the small quantities are deviations from the reference limit cycle. Here, we pursued a bona-fide power expansion in terms of the coupling constant ǫ, and the result differs from the one obtained following [12,53]. In passing, we mention that instead of assuming r j = r j (θ 1 , θ 2 , .…”

“…Apart from the idea of invoking hypothetical three-body interacting limit-cycle oscillators [16], multi-body phase interactions naturally arise if the coupling is nonlinear [17], see also [18]. Instead, for linear pairwise coupling, three-body interactions are a distinctive element of second-order phase approximations, as recently highlighted in [12]. Recognizing the ubiquity of multi-body interactions may also be important for reconstructing phase interactions from data [19].…”

Semiclassical phase reduction theory for quantum synchronization

“…Recent progress in experimental studies has revealed that synchronization can take place in coupled nonlinear oscillators with intrinsically quantum-mechanical origins, such as micro and nanomechanical oscillators [16][17][18][19][20], spin torque oscillators [21], and cooled atomic ensembles [22,23]. Moreover, theoretical studies have been performed on the synchronization of nonlinear oscillators which explicitly show quantum signatures , such as optomechanical oscillators [24][25][26], cooled atomic ensembles [27,28], trapped ions [29][30][31], spins [32], and superconducting circuits [33].…”