In this paper, the synchronization in a hyper-network of coupled dynamical systems is investigated for the first time. An evolving hyper-network model is proposed for better describing some complex systems. A concept of joint degree is introduced, an
d the evolving mechanism of hyper-network is given with respect to the joint degree. The hyper-degree distribution of the proposed evolving hyper-network is derived based on a rate equation method and obeys a power law distribution. Furthermore, the synchronization in a hyper-network of coupled dynamical systems is investigated for the first time. By calculating the joint degree matrix, several simple yet useful synchronization criteria are obtained and illustrated by several numerical examples.
A large deviation principle is derived for stochastic partial differential equations with slow-fast components. The result shows that the rate function is exactly that of the averaged equation plus the fluctuating deviation which is a stochastic part
ial differential equation with small Gaussian perturbation. This also confirms the effectiveness of the approximation of the averaged equation plus the fluctuating deviation to the slow-fast stochastic partial differential equations.
A dynamical system driven by non-Gaussian Levy noises of small intensity is considered. The first exit time of solution orbits from a bounded neighborhood of an attracting equilibrium state is estimated. For a class of non-Gaussian Levy noises, it is
shown that the mean exit time is asymptotically faster than exponential (the well-known Gaussian Brownian noise case) but slower than polynomial (the stable Levy noise case), in terms of the reciprocal of the small noise intensity.
