Pathwise synchronization of global coupled system with linear multiplicative rough noise

This paper focuses on achieving pathwise synchronization in stochastic differential equations with linear multiplicative rough noises, which are fractional Brownian rough paths with Hurst parameter H is an element of (1/3, 1/2). Using rough paths theory, a useful transformation is introduced to convert the equations into random differential equations. Stability and dynamical behavior of the solutions to the equations are discussed, and pathwise synchronization of the solutions to the coupled system is proven. Also we have verified the synchronization results in H & ouml;lder space. And at the end, two alternative forms of noises are considered, and synchronization results are presented. Moreover, numerical simulations are provided to illustrate the results.