Stability of Synchronized Motion in Complex Networks

TL;DR

The work develops a rigorous, self-contained framework for stability of synchronized motion in networks of diffusively coupled identical oscillators. By separating dynamics from network structure via the Laplacian spectrum and employing uniform contraction theory, it derives a clear global synchronization criterion: α > α_c/λ2, with η = αλ2 − α_c governing transient decay. The approach extends naturally to cluster synchronization and hypernetworks, providing reduced stability problems on projected graphs and explicit examples (Lorenz, small-node networks). This perspective yields robust synchronization results with explicit dependence on oscillator dynamics and network connectivity, while connecting to and extending Pecora–Carroll style Lyapunov analyses. The treatment also lays groundwork for generalizations to higher-order interactions and symmetry-driven manifolds, with practical insights for designing synchronizable networks.

Abstract

We give a succinct and self-contained description of the synchronized motion on networks of mutually coupled oscillators. Usually, the stability criterion for the stability of synchronized motion is obtained in terms of Lyapunov exponents. We consider the fully diffusive case which is amenable to treatment in terms of uniform contractions. This approach provides a rigorous, yet clear and concise, way to the important results.

Figures (7)

• Figure 1: Examples of undirected a) and b) and directed c) graphs. The diameter of graph a) is $d= 2$, hence, the graph is connected. The graph b) is disconnected, there is no path connecting the nodes 1 and 2 to the remaining nodes, the diameter is $d = \infty$. However, the graph has two connected components, the upper (1,2) with diameter $d=1$, and the lower nodes (3,4,5) with diameter $d = 2$. Graph c) is directed; the arrow tells the direction of the connection, so node 1 is reachable from node 2, but not the other way around.
• Figure 2: Networks containing four nodes. Their adjacency and Laplacian matrices are represented by $\textbf{\em A}$ and $\textbf{\em L}$. Further details can be found in Table \ref{['table']}.
• Figure 3: Some examples of complex networks.
• Figure 4: The trajectories of the Lorenz system eventually enter an absorbing domain and accumulate on a chaotic attractor. This projection of an attractor resembles a butterfly -- the common name of the Lorenz attractor.
• Figure 5: Two distinct simulations of the time series $x(t)$ and $\tilde{x}(t)$ of the Lorentz systems. The difference between the trajectories is of $0.01$, however this small difference grows with time until a point where the difference is as large as the attractor itself.

Theorems & Definitions (51)

• Theorem 1
• Theorem 2
• Theorem 3: Lyapunov
• Proposition 1
• Proposition 2
• Definition 1: Stability in the sense of Lyapunov
• Definition 2: Asymptotic stability
• Definition 3: Uniform asymptotic stability
• Definition 4
• Example 1