The networks considered are described by fractional derivatives of variable order and are affected by proportional delays, making their synchronization analysis mathematically demanding. The researchers also allowed the system parameters to be uncertain rather than exactly known.

To address this challenge, the team proposed a quantized controller that incorporates linear error states together with a non-smooth switching term. By leveraging this controller, global polynomial synchronization is attained.

The synchronization result was established by constructing an appropriate Lyapunov functional and applying suitable inequality-based techniques. This analytical route allowed the authors to prove that the mismatch between the paired network states converges in the polynomial sense under the proposed control law.

The usefulness of the proposed synchronization approach was illustrated through two computational experiments, which demonstrated the method in simulated settings. As a theoretical and numerical contribution, the study provides a design tool for controlling this class of delayed, uncertain fractional variable-order BAM neural networks rather than a demonstration on physical hardware, and the reported results are supported by the authors’ mathematical analysis and the two presented examples.

Subject of Research: Global polynomial synchronization of uncertain fractional variable-order BAM neural networks with proportional delays via quantized control

Article Title: Global polynomial synchronization of uncertain fractional variable-order BAM neural networks with proportional delay via quantized control

Article References: Pavithra, G., Dharani, S., & Sivakumar, M. (2026). Global polynomial synchronization of uncertain fractional variable-order BAM neural networks with proportional delay via quantized control. Scientific Reports. https://doi.org/10.1038/s41598-026-76904-z

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DOI: 10.1038/s41598-026-76904-z

Keywords: synchronization, fractional-order systems, variable-order derivatives, BAM neural networks, quantized control, proportional delay, Lyapunov functional, stability analysis, uncertain parameters, control theory, Global, polynomial