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Exponential stability of homogeneous positive switched coupled differential–difference systems

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  • Liu, Hui
  • Zhang, Yongping
  • Tian, Dadong

Abstract

While the stability theory of coupled differential–difference systems (CDDSs) is relatively well developed, research on switched CDDSs (SCDDSs) remains underexplored. Notably, SCDDSs are crucial for practical scenarios involving dynamic adjustments, such as epidemic prevention and engineering control systems. This work investigates the stability of positive homogeneous SCDDSs with time-varying delays, addressing a long-standing problem of both theoretical and practical significance. Different from existing studies on homogeneous systems with a dilation map, this paper focuses on a more general class of nonlinear SCDDSs with two dilation maps. Using the piecewise maximum Lyapunov function method, the homogeneity principle, and theoretical tools from positive system analysis, we derive the exponential stability conditions for SCDDSs under average dwell-time (ADT) switching. Compared with asymptotic stability, exponential stability can significantly enhance convergence performance and dynamic response speed. We also present a verifiable sufficient condition for global exponential stability (GES) under the standard dilation map. Furthermore, we establish an explicit expression that allows us to quantify the effects of delay and homogeneity on the decay rate, and identify the optimal decay rate of the system. Subsequently, we extend these results to linear SCDDSs, proving that such systems achieve GES under ADT switching. Finally, the theoretical results are validated by numerical simulations and further applied to a multi-community Susceptible–Infected–Removed (SIR) epidemic model.

Suggested Citation

  • Liu, Hui & Zhang, Yongping & Tian, Dadong, 2026. "Exponential stability of homogeneous positive switched coupled differential–difference systems," Chaos, Solitons & Fractals, Elsevier, vol. 203(C).
    • Handle: RePEc:eee:chsofr:v:203:y:2026:i:c:s0960077925016613
    DOI: 10.1016/j.chaos.2025.117648
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    References listed on IDEAS

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    1. Sun, Yuangong & Tian, Yazhou, 2022. "Polynomial stability of positive switching homogeneous systems with different degrees," Applied Mathematics and Computation, Elsevier, vol. 414(C).
    2. Su, Tan & Kao, Yonggui & Jiang, Daqing, 2025. "Dynamical behaviors of a stochastic SIR epidemic model with reaction–diffusion and spatially heterogeneous transmission rate," Chaos, Solitons & Fractals, Elsevier, vol. 195(C).
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