Hybrid impulses for almost sure quasi-synchronization of stochastic complex networks: an indefinite Lyapunov function method

Abstract.

This article studies quasi-synchronization of stochastic complex networks under hybrid impulses. With parameter mismatches, different from the previous quasi-synchronization results in the sense of mean square on stochastic complex networks, almost sure quasi-synchronization is investigated, in which noises have a positive effect on the synchronization. By utilizing the Lyapunov method, stochastic analysis theory, average impulsive interval, and average impulsive gain, we formulate the almost sure quasi-synchronization criteria when the average impulsive interval z₀ satisfies $z_0<+\mathrm{\infty }$ and $z_0=+\mathrm{\infty }$, respectively, where the synchronizing impulses and the desynchronizing impulses are simultaneously considered. It is worth noticing that in our criteria of quasi-synchronization, the mutual restraints among the average impulsive interval, average impulsive gain, and noise intensity are given. Moreover, for the criteria in this article, the piecewise continuous scalar functions cannot only replace the constant coefficient of the upper bound estimation for the diffusion operator of a Lyapunov function in available literature but also even be unbounded, which has wider applications than some existing works. Our corollaries regarding complete synchronization and analyses of the synchronization for stochastic complex networks with aperiodic intermittent noises provide sufficient conditions in closed form. Finally, the theoretical results are applied to a coupled chaotic Lorentz system, and several numerical examples are presented to demonstrate the advantages of the theoretical results.

Keywords:

• Almost sure quasi-synchronization
• hybrid impulses
• indefinite Lyapunov functions
• stochastic complex networks

Acknowledgments

The authors really appreciate the valuable comments of the editors and reviewers.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by Natural Science Foundation of Shandong Province (Nos. ZR2021MF016, ZR2021MA065 and ZR2021MF032).