What conditions ensure that nonlinear systems will synchronize?

Coupled oscillators exist everywhere – from nature, such as in fireflies that flash synchronously, to electronics, such as in superconducting Josephson junctions where voltage oscillates. Coupled oscillators often synchronize spontaneously, but conditions that guarantee global synchronization are not well-known. To help understand the critical connectivity needed for a network of oscillators to reach synchrony, Townsend et al. investigate the networks of coupled oscillators using a simplified model.

Using their model, the group found evidence that the critical connectivity for guaranteed synchrony is 75%. In other words, if every oscillator in the system is coupled to at least 75% of the others, the system should eventually begin to flash in unison. While the exact value remains unknown, previous work indicates it is in the range of 68-79%.

“I like to think of this as each oscillator is doing its best to flash in unison by watching its friendship group,” said author Alex Townsend.

To investigate the value of the critical connectivity, the authors constructed a family of circulant networks, with each member of the network sharing the same pattern of connections as its peers. By comparing the required number of connections in different sized sets, they determined the percentage point at which a small tweak in the system is enough to change whether it will synchronize.

Real-life biological or chemical systems, like fireflies, have additional intricacies not considered here, and a realistic model would be much more complicated. However, this model serves as a good starting point.

“If we cannot understand synchronization in this simple model, we cannot hope to understand it in any real system,” said author Steven Strogatz.

Source: “Dense networks that do not synchronize and sparse ones that do,” by Alex Townsend, Michael Stillman, and Steven H. Strogatz, Chaos (2020). The article can be accessed at https://doi.org/10.1063/5.0018322 .