Dynamic event-triggered approach for practical fixed-time cluster synchronization control of multilayer complex networks under denial-of-service attacks

Ling Liu; Ling Liu

doi:10.3934/math.2026322

AIMS Mathematics

2026, Volume 11, Issue 3: 7821-7844. doi: 10.3934/math.2026322

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Research article

Dynamic event-triggered approach for practical fixed-time cluster synchronization control of multilayer complex networks under denial-of-service attacks

Ling Liu ^,

School of Mathematics and Systems Science, Guangdong Polytechnic Normal University, Guangzhou 510000, China

Received: 07 January 2026 Revised: 06 March 2026 Accepted: 11 March 2026 Published: 24 March 2026
MSC : 37M05, 37M25

This paper is concerned with the practical fixed-time cluster synchronization control problem of a class of multilayer complex networks (MCNs) with different nonlinearly coupled nodes in different layers under denial of service (DoS) attacks. Each node in the MCNs is modeled by a nonlinear dynamic system. The central aim is to make the error between each cluster of nodes to its own reference trajectory converge to a bounded region in fixed time, while simultaneously achieving communication efficiency for each node. Toward this aim, a distributed dynamic event-triggered mechanism resilient to DoS attacks is proposed such that each node can make its own decisions to transmit (or not) its data of interest over the communication channel. Second, by suitably modeling the DoS attacks, event-based cluster synchronization controllers are constructed that incorporate the dual effects via two core design strategies: Setting the control input to zero during the active DoS attack intervals and adopting a zero-order hold strategy to keep the control input constant with the latest transmitted state data during the inter event intervals of the event-triggered mechanism. Sufficient conditions ensuring the practical fixed-time cluster synchronization of the MCNs under DoS attacks are established by constructing some appropriate Lyapunov functionals. Finally, an illustrative example is presented to validate the effectiveness of the main theoretical results.
- multilayer complex networks,
- cluster synchronization,
- practical fixed-time,
- DoS attacks,
- event-triggered control,
- nonlinear couplings,
- cyber-physical security
Citation: Ling Liu. Dynamic event-triggered approach for practical fixed-time cluster synchronization control of multilayer complex networks under denial-of-service attacks[J]. AIMS Mathematics, 2026, 11(3): 7821-7844. doi: 10.3934/math.2026322

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Abstract

This paper is concerned with the practical fixed-time cluster synchronization control problem of a class of multilayer complex networks (MCNs) with different nonlinearly coupled nodes in different layers under denial of service (DoS) attacks. Each node in the MCNs is modeled by a nonlinear dynamic system. The central aim is to make the error between each cluster of nodes to its own reference trajectory converge to a bounded region in fixed time, while simultaneously achieving communication efficiency for each node. Toward this aim, a distributed dynamic event-triggered mechanism resilient to DoS attacks is proposed such that each node can make its own decisions to transmit (or not) its data of interest over the communication channel. Second, by suitably modeling the DoS attacks, event-based cluster synchronization controllers are constructed that incorporate the dual effects via two core design strategies: Setting the control input to zero during the active DoS attack intervals and adopting a zero-order hold strategy to keep the control input constant with the latest transmitted state data during the inter event intervals of the event-triggered mechanism. Sufficient conditions ensuring the practical fixed-time cluster synchronization of the MCNs under DoS attacks are established by constructing some appropriate Lyapunov functionals. Finally, an illustrative example is presented to validate the effectiveness of the main theoretical results.

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