Generalized outer synchronization and parameters identification of uncertain delayed hypernetworks with stochastic noises

Synchronization of hypernetworks has been extensively researched due to its potential applications in various contexts, such as secure communications and power grid engineering. However, the existing works thus far have primarily been devoted to inner synchronization or complete outer synchronization with identical node dynamics and topology structures. To investigate harmonious coexistence between two networks with nonidentical node dynamics and reduce control costs as well, this paper focuses on generalized outer synchronization (GOS) and parameters identification of uncertain delayed hypernetworks with stochastic noises. Regardless of whether the node parameters are known or not, based on Lyapunov stability and the LaSalle-type invariance principle for stochastic differential equations with multiple delays, some sufficient conditions are exhibited to reach GOS almost surely via adaptive control schemes. Moreover, the unknown node parameter vectors are identified by the designed updating laws. In addition, some corollaries are deduced and indicate that our proposed methods contain single-layer networks and complete synchronization methods as special cases. Furthermore, numerical simulations are provided to verify the effectiveness of our approaches. In particular, it is revealed that network parameters, including time delays, coupling strengths, and noise intensities, have an impact on the performance of the proposed algorithms.