A Lyapunov-Based Distri buted Framework for Complete and Phase Synchronization in Chaotic Multi-Agent Systems

TL;DR

The paper addresses synchronization of chaotic multi-agent systems in a leader-follower network under limited topology information. It introduces a distributed nonlinear coupling combined with Lyapunov stability and matrix-measure analysis to derive tractable sufficient conditions for complete and phase synchronization, while also ensuring robustness to communication delays. Key contributions include explicit Lyapunov-based criteria and a delay-robust Lyapunov-Krasovskii functional, demonstrating faster convergence and lower computational burden than traditional LMI-based and adaptive neural methods. Validation on Lü, Rössler, and Chen systems shows reliable synchronization with minimal steady-state error, even under delays and switching topologies, with potential impact on secure communications and coordinated chaotic dynamics.

Abstract

This paper presents a distributed Lyapunov-based control framework for achieving both complete and phase synchronization in a class of leader-follower multi-agent systems composed of identical chaotic agents. The proposed approach introduces a novel nonlinear coupling mechanism and utilizes Lyapunov stability theory combined with matrix measure analysis to derive explicit synchronization conditions. In contrast to traditional LMI-based or adaptive methods, the present approach guarantees synchronization under limited topological information and reduced computational complexity. Three classical chaotic systems - Roessler, Lu, and Chen - are used to validate the theoretical results, confirming the superior convergence rate and robustness of the proposed scheme.

Figures (8)

• Figure 1: (a) State trajectories of the leader and followers for the Lü system, showing convergence. (b) Synchronization error $E(t)$ decaying to near zero.
• Figure 2: (a) State trajectories of the leader and followers for the Rössler system under communication delay. (b) Synchronization error $E(t)$ showing convergence despite $\tau = 0.5s$ delay.
• Figure 3: Chen system: (a) State trajectories showing phase synchronization with constant offsets. (b) Synchronization errors converging to non-zero steady-state values consistent with theoretical predictions.
• Figure 4: Phase portrait of the Chen leader system, displaying the characteristic double-scroll chaotic attractor in coordinate planes and 3D.
• Figure 5: Eigenvalue distribution of the extended Jacobian for (a) $\alpha = 0.5$ (unstable), (b) $\alpha = 0.8$ (marginally stable), and (c) $\alpha = 0.95$ (asymptotically stable).