Synchronization of nonlinearly coupled networks based on circle criterion

Abstract

The problem of synchronization in networks of linear systems with nonlinear diffusive coupling and a connected undirected graph is studied. By means of a coordinate transformation, the system is reduced to the form of mean-field dynamics and a synchronization-error system. The network synchronization conditions are established based on the stability conditions of the synchronization-error system obtained using the circle criterion, and the results are used to derive the condition for synchronization in a network of neural-mass-model populations with a connected undirected graph. Simulation examples are presented to illustrate the obtained results.

Synchronization in Networks with Nonlinearly Delayed Couplings on Example of Neural Mass Model

“…Note that while T → 0 and choosing ν 0 → 0 such that T 2 /ν 0 → 0, we get the similar conditions of network synchronization as in Plotnikov and Fradkov (2021).…”

“…The condition (3) of Theorem 2 is the same as for the case without delays: this fact follows from the conditions of Circle criterion. This condition was previously checked in Plotnikov and Fradkov (2021): If α > 0, then the matrix Ψ in the condition (3) of the Theorem 2 is Hurwitz.…”

“…To obtain the network equations in new coordinates suppose that the graph of considered network is connected and undirected. Following the steps described in Plotnikov and Fradkov (2021) up to a time-delay one can present the equations of the network (4), (5) in the following form:…”

“…The transfer function of the linear part of the system (7b) can be calculated just like in Plotnikov and Fradkov (2021):…”

“…This paper continuous the work started in Plotnikov and Fradkov (2021). Here synchronization in heterogeneous networks of linear systems with nonlinear delayed diffusive coupling is considered.…”