Coupling Induced Stabilization of Network Dynamical Systems and Switching

TL;DR

The paper addresses stability in diffusively coupled network dynamical systems and how coupling strength and network topology can stabilize or destabilize the network, even under arbitrary switching. It employs Lyapunov methods to derive critical coupling values and sufficient conditions for asymptotic stability, including the existence of common Lyapunov functions for switched networks. The authors show that coupling can stabilize networks of unstable nodes when the coupling spectrum avoids the origin, and they relate stabilizability to graph properties such as bipartiteness. Numerical simulations with four-node Sprott systems corroborate the theoretical results, illustrating both diffusion- and signless-diffusive coupling effects and the impact of switching. The work contributes actionable criteria for designing coupling in complex networks to ensure resilience, with potential extensions to hypernetworks future directions.

Abstract

This paper investigates the stability and stabilization of diffusively coupled network dynamical systems. We leverage Lyapunov methods to analyze the role of coupling in stabilizing or destabilizing network systems. We derive critical coupling parameter values for stability and provide sufficient conditions for asymptotic stability under arbitrary switching scenarios, thus highlighting the impact of both coupling strength and network topology on the stability analysis of such systems. Our theoretical results are supported by numerical simulations.

Figures (8)

• Figure 1: Four diffusively coupled stable Sprott systems sprott2010elegant with parameter $\mu = 0.55$ on the bidirectional graph shown in Fig. \ref{['fig:BidirectionGrap-on-4-nodes']}. The coupling strength is set to $\alpha = \alpha_c - \frac{1}{1000} = 0.0115$, where $\alpha_c = 0.0125$. The resulting network system is stable as predicted by Proposition \ref{['prop:Stability-Interval_NtwkDyn']}. The top left, top right and bottom left plots show trajectories of the 1st, 2nd and 3rd channel respectively of each of the $4$ nodes. The bottom right plot shows the trajectory of the node dynamics in 3D space. The purple dash-dotted line shows the beginning segment of the trajectory to give a sense of orientation. The join state space is $12$-dimensional.
• Figure 2: Same setup as in Fig. \ref{['fig:Stable_Sprott__sub_critical']}., but here the coupling strength is set to $\alpha = \alpha_c + \frac{1}{1000} = 0.0135$. The resulting network system is unstable as predicted by Proposition \ref{['prop:Stability-Interval_NtwkDyn']}.
• Figure 3: Bidirectional graph on a set of four nodes.
• Figure 4: Four signless-diffusively coupled unstable Sprott systems sprott2010elegant with parameter $\mu = 0$ on the bidirectional graph shown in Fig. \ref{['fig:BidirectionGrap-on-4-nodes']}. The coupling parameter is set to $\alpha = \alpha_c - \frac{1}{1000} = -0.6555$, where $\alpha_c = -0.6545$. The resulting network system is stable as predicted by Proposition \ref{['prop:Critical_Stabilizing_coupling_val']}. The top left, top right and bottom left plots show trajectories of the 1st, 2nd and 3rd channel respectively of each of the $4$ nodes. The bottom right plot shows the trajectory of the node dynamics in 3D space. The purple dash-dotted line shows the beginning segment of the trajectory to give a sense of orientation. The join state space is $12$-dimensional. Note that the node dynamics in this regime (i.e. frictionless Sprott system with $\mu = 0$) is in fact chaoticsprott2010elegant, yet with the right coupling we're able to tame it down and effectively stabilize the whole network system.
• Figure 5: Same setup as in Fig. \ref{['fig:Unstable_Sprott__sub_critical']}. Here however, the coupling parameter is set to $\alpha = \alpha_c + \frac{1}{1000} = -0.6535$. The resulting network system is unstable as predicted by Proposition \ref{['prop:Critical_Stabilizing_coupling_val']}. The trajectories are hinting that the dynamics of each channel is evolving on a toroidal manifold, so it's either periodic or quasi-periodic.

Theorems & Definitions (30)

• Proposition 1
• proof
• Proposition 2
• proof
• Definition 1
• Corollary 1
• Remark 1
• Proposition 3
• proof
• Proposition 4