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Critical curves and stability region in a complex network with delayed feedback control

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  • Yang, Ping
  • Lin, Yiping

Abstract

In this paper, a two-dimensional complex network with delayed feedback control is proposed and discussed. For the network without control, the equilibrium of the network is unstable, but under the control with even very small parameters, the equilibrium of the network will become stable. A detailed stability analysis on the controlled system is provided. Differing from the previous works, on the coordinate plane of two delays, the stability region is surrounded by the critical curves. The supercritical and the subcritical Hopf bifurcations are discussed particularly along the boundary of the stability region. Numerical simulations are provided to illustrate the results. The investigation shows that there is stable region surrounded by five critical curves and there exist doubling-periodic solutions and chaotic solutions in the controlled network when the parameters keep away from the stability region. This work provides a theoretical basis for the further application of complex networks.

Suggested Citation

  • Yang, Ping & Lin, Yiping, 2026. "Critical curves and stability region in a complex network with delayed feedback control," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 241(PA), pages 548-561.
    • Handle: RePEc:eee:matcom:v:241:y:2026:i:pa:p:548-561
    DOI: 10.1016/j.matcom.2025.09.011
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    References listed on IDEAS

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    1. M. E. J. Newman & D. J. Watts, 1999. "Scaling and Percolation in the Small-World Network Model," Working Papers 99-05-034, Santa Fe Institute.
    2. Franz Kaiser & Vito Latora & Dirk Witthaut, 2021. "Network isolators inhibit failure spreading in complex networks," Nature Communications, Nature, vol. 12(1), pages 1-9, December.
    3. Iván Y Fernández-Rosales & Larry S Liebovitch & Lev Guzmán-Vargas, 2015. "The Dynamic Consequences of Cooperation and Competition in Small-World Networks," PLOS ONE, Public Library of Science, vol. 10(4), pages 1-13, April.
    4. Duncan J. Watts & Steven H. Strogatz, 1998. "Collective dynamics of ‘small-world’ networks," Nature, Nature, vol. 393(6684), pages 440-442, June.
    5. Wang, Zidong & Lauria, Stanislao & Fang, Jian’an & Liu, Xiaohui, 2007. "Exponential stability of uncertain stochastic neural networks with mixed time-delays," Chaos, Solitons & Fractals, Elsevier, vol. 32(1), pages 62-72.
    6. Zheng, Qianqian & Shen, Jianwei & Pandey, Vikas & Guan, Linan & Guo, Yantao, 2023. "Turing instability in a network-organized epidemic model with delay," Chaos, Solitons & Fractals, Elsevier, vol. 168(C).
    7. Li, Chunguang & Chen, Guanrong, 2004. "Phase synchronization in small-world networks of chaotic oscillators," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 341(C), pages 73-79.
    8. Gerardo Iñiguez & Sara Heydari & János Kertész & Jari Saramäki, 2023. "Universal patterns in egocentric communication networks," Nature Communications, Nature, vol. 14(1), pages 1-9, December.
    9. Chang Liu & Fengli Xu & Chen Gao & Zhaocheng Wang & Yong Li & Jianxi Gao, 2024. "Deep learning resilience inference for complex networked systems," Nature Communications, Nature, vol. 15(1), pages 1-16, December.
    10. Zhou, Jiaying & Ye, Yong & Arenas, Alex & Gómez, Sergio & Zhao, Yi, 2023. "Pattern formation and bifurcation analysis of delay induced fractional-order epidemic spreading on networks," Chaos, Solitons & Fractals, Elsevier, vol. 174(C).
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