# Research: Synchronization of oscillators on a network

Birds can fly because they flap their wings in-phase at the *same time*. We can walk because our legs move out-of-phase at the *same time*. At a concert, people move to the rhythm, some left to right, some front to back, others in whichever way, but everyone is in “*sync*” with each other and the music. This phenomenon of two or more bodies performing some action at the same time in coordination is called *Synchronization*.

Synchronization can be observed in a wide variety of systems, from the flight of birds to the dancing people in a concert, to the operation of a network of computers. Therefore understanding this single phenomenon can give us an insight into a range of completely unrelated systems — making it a fascinating field of study (broadly, nonlinear dynamics).

One of our researchers, Dr Kaustubh Manchanda, works in the general area of nonlinear dynamics. Below is the summary of their recently published work [1] on the synchronization of oscillators in a network, in the author’s words:

In this work, we consider a star shaped graph (see Fig. 1). The node sitting in the center (h) is called a hub and here it is connected to eight edges: hence its degree of connectivity is eight. The nodes labelled ℗ are called peripheral nodes. It’s evident from the figure that each one of them has degree of connectivity as one. Examples of star networks can be drawn from technological systems. For instance, the design of the star network comes from the telecommunication system, where the central switching station manages all calls. Star network is ubiquitous in computer networking. In a local area network (LAN), all nodes (workstations or other devices) are directly connected to a common central computer (hub). One disadvantage of a star network is that when the hub fails, the whole system can become non-functional.

On each of these nodes, we place a Kuramoto oscillator [2]. Simply put, a Kuramoto oscillator is an oscillator which has one degree of freedom namely its phase. These oscillators are commonly used in models of chemical oscillations as well as in the areas of neuroscience. Each oscillator has an intrinsic frequency. ($$\omega_h$$: hub, $$\omega_p$$: peripheral). The peripheral nodes interact with each other via the hub, this can be likened to a situation where several pendula are connected to the same beam and their interactions are mediated via the vibrations of the molecules of the beam. In our work, we assume a positive degree-frequency correlation which means nodes which have a higher degree (in our case it’s the hub), the oscillator placed on the hub has a higher intrinsic frequency as compared to the peripheral oscillators. This assumption aligns with commonly found star networks around us. The typical number of oscillators that we connected was 60. Another important thing to consider on such networks is the speed at which information travels between nodes, a realistic scenario is finite speed of propagation leading to time delay in this network.

The question we address in this work is about the onset of coherence or synchronization in this network of oscillators. We are interested in computing the threshold as well the route to synchrony, namely how the different oscillators organize themselves into the state of coherence. Our main finding was that the onset of synchronization via remote synchrony [3] occurred through continuous (first-order) and discontinuous (second order) phase transitions depending on the value of delay. For the positive degree-frequency correlated star network of phase oscillators, the time period of the system induced by average frequency is given by $$T_{av} = \dfrac{\pi}{\omega_p}$$. Delays in the first half of the time period give continuous first-order phase transitions. In contrast, delays in the second half of the time period give discontinuous second-order phase transitions. At the critical point, the system undergoes a saddle node bifurcation.

Knowing how synchronization arises in the coupled dynamical system gives us the insight into the mechanisms to obtain desirable behaviour and control undesirable dynamics. A star network of phase oscillators is a simple and essential system for understanding the onset of synchronization dynamics. Such coherent dynamics is crucial to understanding how coupled dynamical systems regulate dynamics optimally.

### Credits:

Introduction in this post written by Sanjana. Article and abstract for *Hands-on Physics* by Kaustubh based on their recent publication [1].

### References:

[1] Kaustubh, Umeshkanta, “*Continuous and explosive synchronization of phase oscillators on star network: Effect of degree-frequency correlations and time-delays*“, https://doi.org/10.1016/j.chaos.2023.113326, 2023.

[2] “Kuramoto Model”, https://en.wikipedia.org/wiki/Kuramoto_model

[3] “Remote Synchronization”, http://www.scholarpedia.org/article/Remote_synchronization