A geometric approach for stability analysis of delay systems: Applications to network dynamics

TL;DR

The paper develops a geometric framework to analyze the stability of linear time-delay systems with complex-valued coefficients, arising from asymmetric and potentially random networks, by introducing stability crossing curves (SCCs) in the complex plane. It proves a root-continuity principle and derives a geometric procedure to construct stability regions Ω from SCC geometry, applicable to both discrete and distributed delays. The approach is demonstrated on scalar DDEs and extended to networked systems via the master stability function, yielding explicit stability criteria for car-following and Kuramoto-like networks, including cases with Gamma delays and random networks via the Circular Law. The work provides a practical toolkit for designing delayed control strategies to achieve consensus or desynchronization, with broad applicability to engineering and neuroscience, and opens avenues for time-varying delays and PDE extensions.

Abstract

Investigating the network stability or synchronization dynamics of multi-agent systems with time delays is of significant importance in numerous real-world applications. Such investigations often rely on solving the transcendental characteristic equations (TCEs) obtained from linearization of the considered systems around specific solutions. While stability results based on the TCEs with real-valued coefficients induced by symmetric networks in time-delayed models have been extensively explored in the literature, there remains a notable gap in stability analysis for the TCEs with complexvalued coefficients arising from asymmetric networked dynamics with time delays. To address this challenge comprehensively, we propose a rigorously geometric approach. By identifying and studying the stability crossing curves in the complex plane, we are able to determine the stability region of these systems. This approach is not only suitable for analyzing the stability of models with discrete time delays but also for models with various types of delays, including distributed time delays. Additionally, it can also handle random networks. We demonstrate the efficacy of this approach in designing delayed control strategies for car-following systems, mechanical systems, and deep brain stimulation modeling, where involved are complex-valued TCEs or/and different types of delays. All these therefore highlight the broad applicability of our approach across diverse domains.

Figures (21)

• Figure 1: According to Theorem \ref{['theorem1']}, all unstable roots of the TCE $F(\lambda,L)=0$ with $L\in\Theta$ are located within a semicircular bounded by the contour $C_R=g_1\cup g_2$ and vary continuously with respect to $L$. From this, it follows that the value of ${\rm NU}(L)$ changes only if a root crosses the imaginary axis $\mathbb{C}_0$, indicating that these changes only occur at the SCCs.
• Figure 2: The SCCs for system \ref{['1105']} (the blue curves) separate the complex plane into several regions. The origin $L=0$, which is highlighted by a red dot, lies within Region $A$. According to Theorem \ref{['theorem1']}, ${\rm NU}(0)=1$ implies ${\rm NU}({\rm Region}~A)=1$.
• Figure 3: The blue curve represents the local SCC $L(\beta)$, the black arrow represents the increasing direction of $\beta$, and $\vec{n}$ represents the normal vector defined in Theorem \ref{['geometric']}. The direction of $\vec{n}$ is obtained by a 90-degree-counterclockwise rotation of the tangent vector $L'(\beta)$. As proved in Theorem \ref{['geometric']}, when $L$ moves from $L^*-\epsilon\vec{n}$ (the brown dot) to $L^*+\epsilon\vec{n}$ (the purple dot), one root of equation $F(\lambda,L)=0$ undergoes a continuous transition from $\mathbb{C}_+$ to $\mathbb{C}_-$, crossing through $\mathbb{C}_0$ at $\lambda^*$ as $L=L^*$ (the black dot). Therefore, ${\rm NU}({\rm Region}~A)-{\rm NU}({\rm Region}~B)=-1$.
• Figure 4: The SCCs $L=L_{k}(\beta)$ for system \ref{['example2']} separate the complex plane into several regions, while the black arrows represent the increasing directions of $\beta$. Here, the red dot represents the origin and from this, we deduce that ${\rm NU}({\rm Region}~A)=1$. The value of ${\rm NU}$ for each region is easily obtained using Theorem \ref{['geometric']}, so that the stability region $\Omega$ for the considered system is ${\rm Region}~G$ (see the green shaded area).
• Figure 5: The directions of the SCCs (the blue curves) are determined by different signs of $\theta'(\beta)$. Here, the black arrows represent the increasing directions of $\beta$. Multiple brown lines, originating from the origin (the red dot), intersect the SCCs at various points. The purple square dot corresponds to $\theta'(\beta^*)<0$, while the magenta triangle dots correspond to $\theta'(\beta^*)>0$. From Theorem \ref{['theorem3']}, ${\rm NU}({\rm Region}~B)-{\rm NU}({\rm Region}~A)=-1$, ${\rm NU}({\rm Region}~C)-{\rm NU}({\rm Region}~A)=1$, and ${\rm NU}({\rm Region}~D)-{\rm NU}({\rm Region}~A)=1$.

Theorems & Definitions (27)

• Remark II.1
• Definition II.2
• Definition II.3
• Definition II.4
• Definition II.5
• Definition II.6
• Definition III.1
• THEOREM III.2
• proof
• Remark III.3