The global structural stability and application of the toroidal differntial equations\left\{ {\begin{array}{*{20}c} {\dot x = a_1 [\sin (x - y) - \sin x] - \Delta \omega } \\ {\dot y = a_2 [\sin (y - x) - \sin y] - \Delta \omega } \\ \end{array} } \right.

In this paper, the global structural stability of the toroidal differential equationshas been obtained, and applied to the cross-coupled phase - locked loop, where Δω≥0     

Dynamics of stochastic non-Newtonian fluids driven by fractional Brownian motion with Hurst parameter H ∈ （1/4,1/2）

A two-dimensional (2D) stochastic incompressible non-Newtonian fluid driven by the genuine cylindrical fractional Brownian motion (FBM) is studied with the Hurst parameter $H \in \left( {\tfrac{1} {4},\tfrac{1} {2}} \right)$ under the Dirichlet boundary condition. The existence and regularity of the stochastic convolution corresponding to the stochastic non-Newtonian fluids are obtained by the estimate on the spectrum of the spatial differential operator and the identity of the infinite double series in the analytic number theory. The existence of the mild solution and the random attractor of a random dynamical system are then obtained for the stochastic non-Newtonian systems with $H \in \left( {\tfrac{1} {2},1} \right)$ without any additional restriction on the parameter H.