Synchronization of Pulse-Coupled Oscillators and Clocks Under Minimal Connectivity Assumptions

Abstract

establish synchronization of oscillators with a delay-advance phase-response curve over strongly connected networks. In this paper we extend this result by relaxing the connectivity condition to the existence of a root node (or a directed spanning tree) in the graph. This condition is also necessary for synchronization.

Synchronization of Kuramoto Oscillators in a Bidirectional Frequency-Dependent Tree Network

“…To simulate the results of the event-triggered algorithm, we first take a star graph with 4 nodes with the same initial conditions as the above examples. We set ω = [20,18,16,6] and κ = 1.1. The results is shown in Figure 6-(1).…”

“…In addition to studying frequency synchronization, we present an event-based algorithm for synchronization in star networks, which is a special case of a tree network, assuming a specified κ which may not necesarily satisfy the sufficient condition for synchronization. Compared with [16], we consider a different underlying dynamics for the oscillatory network and design a different algorithm. This paper is organized as follows.…”

On the global synchronization of pulse-coupled oscillators interacting on chain and directed tree graphs

“…From the lower plot of Fig. 4, we can also see that the length of the shortest containing arc V c , which is widely used as a Lyapunov function in local synchronization analysis [24,29,32,34], is not appropriate for global PCO synchronization as it may not decrease monotonically. Along the same line, the firing order which is invariant in [4,28,42], and [32], is not constant in the considered dynamics as exemplified in Then we considered N = 10 PCOs on a directed tree graph, as illustrated in Fig.…”

“…Our main focus is on the global synchronization of PCOs under undirected chain graphs, but the results are easily extendable to PCO synchronization under directed chain/tree graphs. Note that the chain or directed tree graphs are basic elements for constructing more complicated graphs and are desirable in engineering applications where reducing the number of connections is important to save energy consumption and cost in deploy- [3, 13-19, 28, 31, 32] [ 11, 12, 20-28, 32, 33] [34] [29,30] Almost global synchronization or synchronization with probability one [2,35,36] [ [37][38][39] Global synchronization Discrete state synchronization [40] [39]…”

Pattern phase diagram of spiking neurons on spatial networks

“…As χ increases, the CCDF extends towards the left -at first forming a power law and eventually a supercritical distribution (bottom rightmost panel). Rows 3 and 6: blue dots correspond to the temporal average of the spatial spectral power S Φ (λ) defined in(2). The solid red line is a truncated power law fit of S Φ (λ), and the purple (dashed) line is the corner wavelength χ, as determined by the fitting method described in Appendix B. try of the model by taking the radial mean of s Φ ( λ):…”