Region of Synchronization Estimation for Complex Networks via SOS Programming

TL;DR

The paper tackles estimating the region of synchronization (ROS) for complex networks with nonlinear node dynamics and a synchronization manifold as the target. It introduces an exponential guidance-barrier function (EGBF) and formulates a sum-of-squares (SOS) programming framework to compute ROS along the synchronization manifold, with an extension to ROS around an equilibrium via a similar SOS approach. The main contributions include Theorem 1, which guarantees that a positive level set of a computed $V(\cdot)$ lies inside the ROS along the manifold, and Theorem 2 for the equilibrium case, both enabling larger, more general-shaped ROS than prior methods. The work is validated with two numerical examples showing effective ROS estimation and convergence of trajectories to the synchronization manifold, highlighting practical applicability to design and analysis of nonlinear networks.

Abstract

In this article, we explore the problem of the region of synchronization (ROS) for complex networks with nonlinear dynamics. Given a pair of state- and target- sets, our goal is to estimate the ROS such that the trajectories originating within it reach the target set (i.e., synchronization manifold), without leaving the state set before the first hitting time. In order to do so, an exponential guidance-barrier function is proposed to construct the ROS along the synchronization manifold, and the corresponding sufficient conditions for estimating the ROS are developed. The resulting conditions lead to a sum-of-squares programming problem, thereby affording a polynomial-time solvability. Furthermore, when the synchronization manifold reduces to an equilibrium point, our method not only estimates a larger ROS compared to existing results but also allows the ROS to take more general shapes. Finally, we present two numerical examples to demonstrate the effectiveness of the theoretical results.

Figures (4)

• Figure 1: Black and red curves -- $\partial \mathcal{X}$ and $\partial \mathcal{X}_{T}$; blue and purple curves -- $\partial \mathcal{R}$ in Example 1.
• Figure 2: The error trajectories of network \ref{['vandepol']}.
• Figure 3: Black and red curves -- $\partial \mathcal{X}$ and $\partial \mathcal{X}_{T}$; blue and purple curves -- $\partial \mathcal{R}$ in Example 2.
• Figure 4: The state trajectories of network \ref{['jet-engines']}.