Stability and bipartite synchronization of fractional-order coupled reaction–diffusion neural networks under unbalanced graph

This paper delves into the analysis of stability and bipartite synchronization (BS) in fractional-order coupled neural networks with reaction–diffusion terms (FORDNNs) under structurally unbalanced signed graphs. Central to our investigation is the novel observation in structurally unbalanced graphs without cycles, where specific nodes under particular structures are capable of achieving BS, while others maintain stability. Subsequently, for unbalanced graphs with negative rooted cycles, we present criteria that guarantee the stability of FORDNNs. Moreover, under structurally balanced graphs, sufficient conditions for achieving BS are established. A novel approach is introduced, wherein synchronization is attained without the presence of a leader node, marking a significant departure from conventional models. Finally, illustrative examples are provided, demonstrating the effectiveness and relevance of our findings. • In unbalanced graphs without cycles, certain nodes achieve BS, others stable • Criteria for stability in unbalanced graphs with negative cycles. • BS achieved in balanced graphs without a leader node.