17 - Stability Analysis of Stochastically Switching Kuramoto Networks

Motivated by real-world networks with evolving connections, we analyze how stochastic switching affects patterns of synchrony and their stability in networks of identical Kuramoto oscillators with inertia. Stochastic dynamical networks are a useful model for many physical, biological, and engineering systems that have evolving topology, but they have proven to be difficult to work with, and the analytical results are rare. These networks have two characteristic time scales, one is associated with intrinsic dynamics of individual oscillators comprising the network, and the other corresponds to switching period of on-off connections. In the limit of fast switching, the relation between the dynamics of the stochastic network and the static network can be obtained by replacing the switching connections with their mean. We use averaging and Lyapunov function methods to elucidate this non-trivial relationship. We prove that clusters of synchrony which stably appear in the averaged static network can also be observed in the original stochastic network and derive bounds on the switching frequency that guarantee, in a probabilistic sense, the convergence to a stable cluster solution. We also demonstrate the emergence and persistence of chimera states in these stochastic networks

Dispersive versus Dissipative Coupling for Frequency Synchronization in Lasers

Coupling-enabled frequency synchronization is essential for an array of light sources operating in a photonic system. Using a two-dimensional nonlinear oscillator model of a laser, we analyze the role of two distinct types of coupling, dispersive and dissipative, in promoting frequency locking between two nonidentical lasers. In both scenarios the two oscillators synchronize into a frequency-locked state when the coupling level exceeds a critical value. We show that the onset of dispersive and dissipative synchronization processes is associated with hard and soft frequency transitions, respectively. Through analysis and numerics, we demonstrate that the dispersive coupling yields bistable synchronization modes, accompanied by asymmetric intensities, and the frequency controlled by the coupling strength. In contrast, dissipative coupling induces monostable synchronization with symmetric intensities and a coupling-independent frequency. Our results are expected to provide a basis for understanding the coupling mechanisms of frequency locking toward controlling synchronization in laser arrays

Spectral Network Principle for Frequency Synchronization in Repulsive Laser Networks

Network synchronization of lasers is critical for reaching high-power levels and for effective optical computing. Yet, the role of network topology for the frequency synchronization of lasers is not well understood. Here, we report our significant progress toward solving this critical problem for networks of heterogeneous laser model oscillators with repulsive coupling. We discover a general approximate principle for predicting the onset of frequency synchronization from the spectral knowledge of a complex matrix representing a combination of the signless Laplacian induced by repulsive coupling and a matrix associated with intrinsic frequency detuning. We show that the gap between the two smallest eigenvalues of the complex matrix generally controls the coupling threshold for frequency synchronization. In stark contrast with Laplacian networks, we demonstrate that local rings and all-to-all networks prevent frequency synchronization, whereas full bipartite networks have optimal synchronization properties. Beyond laser models, we show that, with a few exceptions, the spectral principle can be applied to repulsive Kuramoto networks. Our results may provide guidelines for optimal designs of scalable laser networks capable of achieving reliable synchronization

Modeling of ultrasound tomographic imaging for non-destructive inspection of underwater structures

In non-destructive inspection regular ultrasound techniques provide detection and estimate size of specific defects or undesired objects in different structures. The method of tomographic imaging modeled in this paper can provide the correct position and shape of the object inside underwater structures to be inspected. The proposed method is based on 2-D full waveform inversion of reflected ultrasound field registered by the array of sensors at a short distance from the target object. The computationally intensive algorithm for tomographic image reconstruction is described and tested with focus on underwater pipeline ultrasound inspection. The results of numerical modeling using parallel calculations and physical modeling are presented and discussed

Emergence of the London Millennium Bridge instability without synchronisation.

The pedestrian-induced instability of the London Millennium Bridge is a widely used example of Kuramoto synchronisation. Yet, reviewing observational, experimental, and modelling evidence, we argue that increased coherence of pedestrians' foot placement is a consequence of, not a cause of the instability. Instead, uncorrelated pedestrians produce positive feedback, through negative damping on average, that can initiate significant lateral bridge vibration over a wide range of natural frequencies. We present a simple general formula that quantifies this effect, and illustrate it through simulation of three mathematical models, including one with strong propensity for synchronisation. Despite subtle effects of gait strategies in determining precise instability thresholds, our results show that average negative damping is always the trigger. More broadly, we describe an alternative to Kuramoto theory for emergence of coherent oscillations in nature; collective contributions from incoherent agents need not cancel, but can provide positive feedback on average, leading to global limit-cycle motion

Dynamics of Stochastically Blinking Systems. Part II: Asymptotic Properties

We study stochastically blinking dynamical systems as in the companion paper (Part I). We analyze the asymptotic properties of the blinking system as time goes to infinity. The trajectories of the averaged and blinking system cannot stick together forever, but the trajectories of the blinking system may converge to an attractor of the averaged system. There are four distinct classes of blinking dynamical systems. Two properties differentiate them: single or multiple attractors of the averaged system and their invariance or noninvariance under the dynamics of the blinking system. In the case of invariance, we prove that the trajectories of the blinking system converge to the attractor(s) of the averaged system with high probability if switching is fast. In the noninvariant single attractor case, the trajectories reach a neighborhood of the attractor rapidly and remain close most of the time with high probability when switching is fast. In the noninvariant multiple attractor case, the trajectory may escape to another attractor with small probability. Using the Lyapunov function method, we derive explicit bounds for these probabilities. Each of the four cases is illustrated by a specific example of a blinking dynamical system. From a probability theory perspective, our results are obtained by directly deriving large deviation bounds. They are more conservative than those derived by using the action functional approach, but they are explicit in the parameters of the blinking system

Synchronization in complex networks with blinking interactions

We propose a new model of small-world networks of cells with a time-varying coupling and study its synchronization properties. In each time interval of length /spl tau/ such a coupling is switched on with probability p and the corresponding switching random variables are independent for different links and for different times. At each moment the coupling corresponds to a small-world graph, but the shortcuts change from time interval to time interval, which is a good model for many real-world dynamical networks. We prove that for the blinking model, a few random shortcut additions significantly lower the synchronization threshold together with the effective characteristic path length. Short interactions between cells, as in the blinking model, are important in practice. To cite prominent examples, computers networked over the Internet interact by sending packets of information, and neurons in our brain interact by sending short pulses, called spikes. The rare interaction between arbitrary nodes in the network greatly facilitates synchronization without loading the network much. In this respect, we believe that it is more efficient than a structure of fixed random connections